D-Modules and Representation TheoryD-MODULES AND REPRESENTATION THEORY 3 1.3. Conventions and notation. We collect here some notations which we will use throughout this essay. We work

D-Modules and Representation Theory

Part III Essay Topic 96

D-MODULES AND REPRESENTATION THEORY

Contents

1. Introduction 11.1. Motivation 11.2. Organization and references 21.3. Conventions and notation 31.4. Acknowledgments 32. D-modules on smooth algebraic varieties 32.1. Definition and examples 32.2. Left and right D-modules 62.3. Pullback and pushforward of D-modules 82.4. The singular support 142.5. Holonomic D-modules and duality 172.6. D-modules on P1 213. The geometric setting 233.1. Preliminaries on semisimple algebraic groups 243.2. Flag and Schubert varieties 243.3. Equivariant vector bundles on the flag variety 243.4. Preliminaries on Lie algebras 263.5. Lie algebra actions on G-equivariant sheaves 263.6. Chevalley and Harish-Chandra isomorphisms 274. Beilinson-Bernstein localization 304.1. Twisted differential operators 304.2. Lie algebroids and enveloping algebras on the flag variety 314.3. A family of twisted differential operators 324.4. The localization functor and the localization theorem 334.5. The map U(g)→ Γ(X,DλX) 354.6. The geometry of the nilpotent cone and Kostant’s Theorem 364.7. The global sections functor on the flag variety 41References 44

1. Introduction

1.1. Motivation. Let X be a smooth algebraic variety over a field of characteristic 0. This essay will beprimarily concerned with D-modules, or modules over a sheaf D of differential operators defined on X.On general varieties, D-modules behave in a much more structured manner than O-modules, and thisoften allows them to carry more global information. For instance, as we shall point out, it is possible torecover the de Rham cohomology of a complex algebraic variety in terms of some D-modules on it. Ourprimary focus in this essay, however, will be on the relationship between the theory of these differentialoperators and representation theory.

Let now G be a semisimple algebraic group over an algebraically closed field. Associated to anysuch G is a semisimple Lie algebra g which we often think of as representing the infinitesimal behaviorof G. Indeed, g can be realized as the space of invariant vector fields on G, and we can attempt tostudy its representation theory from this perspective. Broadly speaking, however, this approach has twodrawbacks. First, this characterization is too global, as the invariance condition does not allow us tounderstand different aspects of the representation theory of g at once.1 We might consider modifyingthis approach to use instead the sheaf of vector fields on G, discarding the invariance condition.

Date: May 6, 2011.1Our precise meaning here is that it does not allow us to study all the Borel subalgebras of g at once.

1

2 D-MODULES AND REPRESENTATION THEORY

This is much closer to the correct notion. However, we must make one further modification; it turns outthat we would like to consider only the behavior of the Borel subalgebras of g simultaneously. Thus, weshould consider only the space of vector fields invariant under the action of a Borel subgroup B ⊂ G. Theresulting sheaf of rings generated by these vector fields is exactly the sheaf of differential operators DXon the flag variety X := G/B of G. The primary result presented in this essay, the Beilinson-Bernsteinlocalization theorem, states “approximately” that we have the following correspondence

DX -modules ↔ U(g)-representations

between DX -modules on the flag variety and representations of the corresponding Lie algebra. We say“approximately,” however, because to make this correspondence rigorous (or true!) requires us to specifyrefinements on both sides.

These twists will result from the essentially different nature of the two objects. As a sheaf, a DX -module contains a significant amount of local information, which in this case will correspond to the actionof the Borel subalgebras of our Lie algebra. On the other hand, U(g)-representations seem are quite globalobjects, as the choice of such a representation often requires a global choice of Borel subalgebra b ⊂ g.It is remarkable, therefore, that such a correspondence can exist. Indeed, the way that it avoids thisseeming contradiction is by on the one hand studying non-trivial cases where the local behavior of allBorel subalgebras coincide (on the D-module case), and on the other hand considering a global invariantthat is independent of the choice of Borel (on the U(g)-representation side). The relevant objects in eachcase will be the characters of the abstract Cartan and the central characters of the center Z(g) of U(g),leaving us with the following refined correspondence, known as the Beilinson-Bernstein correspondence.

“DX -modules” with specified b-action for all b ↔ U(g)-representations with specific Z(g)-action

The purpose of this essay is to give an introduction to the theory of D-modules and then to rigorouslyformulate and prove this correspondence.

1.2. Organization and references. Let us now discuss the specific structure of this essay. In Section2, we give an introduction to the theory of D-modules, highlighting the central results of Kashiwara’sTheorem (Theorem 2.19), Bernstein’s Inequality (Theorem 2.32), and the b-function lemma (Lemma2.43).

In Section 3, we describe the flag variety of a semisimple algebraic group and characterize its rela-tionship with the corresponding Lie algebra. Of particular importance will be the statements and conse-quences of the Chevalley and Harish-Chandra isomorphisms (Theorems 3.13 and 3.16, respectively), asthey will feature prominently in the statement of the Beilinson-Bernstein correspondence.

In Section 4, our main work takes place; we formulate and prove the Beilinson-Bernstein correspon-dence. After formulating the statement in terms of a family of sheaves DλX of twisted differential opera-tors, there are two main steps. First, we show that the global sections Γ(X,DλX) correspond to quotientsof the universal enveloping algebra U(g) by the kernels of central characters. The key input for this stepwill be a close study of the geometry of the nilpotent cone via its Springer resolution. To finish, we showthat the global sections functor defines the desired equivalence of categories. The approach we follow,which is the original one taken by Beilinson and Bernstein, proceeds by using the additional informationprovided by the differential structure to relate any given DλX -module to one where the desired resultholds.

The material we present here is not original, but rather drawn from a number of different sources,detailed as follows. For the theory of D-modules, we are generally indebted to [Ber83] and [HTT08],which we have referenced throughout, and were extremely important in our study of the subject. We alsoconsulted [BGK+87] and [Gai05] to much lesser extents. For the proof of the equivalence of categories inthe Beilinson-Bernstein correspondence, we present essentially the original proof given in [BB81], thoughour treatment benefited greatly from some other viewpoints on the proof, particularly those of [Kas89],[HTT08], and [Gai05]. We also referred to [BB93], [BB83], [Beı83], and [BK81] for some context aboutthe ideas involved. For the geometry of the nilpotent cone, we followed mainly the treatment of [Gai05],consulting also [HTT08] and the original papers of [Kos59] and [BL96]. We have assumed backgroundknowledge from algebraic geometry, algebraic groups, and Lie algebras throughout. We attempt, however,to provide relevant citations (to a perhaps excessive degree). The sources for these are mentioned intext. Finally, we have attempted to be thorough in our exposition, but we have throughout omitted theproofs of some facts which would have taken us too far afield.

D-MODULES AND REPRESENTATION THEORY 3

1.3. Conventions and notation. We collect here some notations which we will use throughout thisessay. We work over a field k of characteristic 0, which beginning in Section 3 we will assume to bealgebraically closed. Denote by Sch/k the category of schemes over k. Let X be a scheme, possiblyequipped with the action of an algebraic group G. Denote by QCoh(X) the category of quasi-coherentsheaves on X and by QCoh(X)G the category of G-equivariant quasi-coherent sheaves on X. By a vectorbundle M on X, we mean a locally free sheaf on X, and we denote by SymX M the geometric realizationof M, which has structure sheaf SymOX M∗. By Hom, we mean the sheaf Hom.

Let X now be a smooth algebraic variety. By TX and T ∗X , we mean its tangent and cotangent bundles,respectively. By ΩX = det(T ∗X), we mean the line bundle of top differential forms on X.

1.4. Acknowledgments. This essay was written for Part III of the Mathematical Tripos in academicyear 2010-2011. I would like to thank Professor Ian Grojnowski for setting this essay and for severalhelpful conversations. I would also like to thank George Boxer for useful discussions in the course oflearning the basic theory of D-modules and Jonathan Wang for several useful discussions relating toderived categories in algebraic geometry.

2. D-modules on smooth algebraic varieties

In this section, we introduce and develop the theory of D-modules on smooth algebraic varieties. Weemphasize the portions of the theory which provide some relevant geometric context for the topic of thesecond half of this essay, the Beilinson-Bernstein localization theorem. The main source for this sectionis the classical notes of Bernstein [Ber83] and the book of [HTT08], supplemented by the notes [Gai05]and the book [BGK+87].

2.1. Definition and examples. Let X be a smooth algebraic variety over a field k of characteristic0. If X = Spec(A) is affine, define a differential operator d of order at most n to be a k-linear mapd : A→ A such that for all f0, . . . , fk ∈ A, we have

[fn, [fn−1, [· · · , [f0, d]]]] = 0

as a k-linear map A→ A, where we view an element f ∈ A as a k-linear map A→ A via multiplication.Such operators form a ring D(A) with multiplication given by composition. We call this the ring ofdifferential operators on Spec(A). Define a filtration FnD(A) ⊂ D(A) on D(A) by letting FnD(A) consistof the differential operators of order at most n. We call this filtration on D(A) the order filtration.

Example 2.1. If A = k[t1, . . . , tn], then D(A) = k[t1, . . . , tn, ∂1, . . . , ∂n] is generated over k by ti andthe formal partial derivatives ∂i subject to the commutation relations

[ti, tj ] = 0, [∂i, ∂j ] = 0, and [∂i, tj ] = δij ,

where δij denotes the Kronecker delta function.

We may glue this notion to a notion of differential operators on the entire variety X. For this, we saythat a sheaf A of non-commutative OX -algebras is quasi-coherent (with respect to the left OX -structure)if for any affine U and f ∈ OU , we have

Γ(Uf ,A) = OUf ⊗OUΓ(U,A).

Of course, when A is commutative, this restricts to the usual notion of quasi-coherence. We say A isquasi-coherent with respect to the right OX -structure if we have

Γ(Uf ,A) = Γ(U,A) ⊗OUOUf

for all affine U and f ∈ OU . We will always mean quasi-coherence with respect to the left structure.

Lemma 2.2. There exists a sheaf DX of OX-algebras quasi-coherent with respect to both the left andright OX-actions such that

Γ(Spec(A),DX) = D(A).

Proof. To check that DX patches correctly into a quasi-coherent sheaf on X, we must check that

D(Af ) ' D(A)⊗AAf ' Af ⊗

AD(A)

for any non-nilpotent f ∈ A. We will only give a sketch of one case here and refer the reader to [Gai05,Proposition 5.5] for the complete proof. Construct the map

φ : D(A)→ D(Af )


inductively on FiD(A) as follows. On F0D(A) = A, it is just the identity map. Suppose now that φ isdefined on FiD(A); for d ∈ Fi+1D(A) and g

fn ∈ Af , we have [fn, d] ∈ FiD(A), motivating the definition

φ(d)

(g

fn

):=

d(g)

fn+φ([fn, d])(g/fn)

fn.

It is clear that this yields a well-defined map φ that is compatible with the filtrations. To check thatit is an isomorphism, it remains only to check that it is an isomorphism on the associated graded level,which will follow from Proposition 2.5 below.2

We call DX the sheaf of differential operators on X. It naturally inherits an order filtration FnDX .The following results show that FnDX gives DX the structure of a sheaf of filtered rings and providesome basic properties.

Lemma 2.3. For non-negative integers n1, n2, we have that

Fn1DX · Fn2DX ⊆ Fn1+n2DX .

Proof. Take d1 ∈ Fn1DX and d2 ∈ Fn2

DX . Then, for any f0, . . . , fn1+n2∈ OX , we note that

[f0, [f1, · · · [fn1+n2 , d1d2]]]

is the sum of monomials of the form

[fi1 , [fi2 , . . . , [fik , d1]]] · [fj1 , [fj2 , . . . , [fjn1+n2+1−k , d2]]],

where i1, . . . , ik and j1, . . . , jn1+n2+1−k form a partition of 0, . . . , n1 + n2. By the definitionof Fn1DX and Fn2DX , this means that either [fi1 , [fi2 , . . . , [fik , d1]]] or [fj1 , [fj2 , . . . , [fjn1+n2+1−k , d2]]]vanishes for each monomial, hence the entire sum vanishes and d1d2 ∈ Fn1+n2

DX as desired.

Proposition 2.4. The order filtration on DX satisfies the following:

(i) F0DX ' OX , and(ii) F1DX ' OX ⊕ TX .

Proof. For (i), notice that local sections ξ ∈ F0DX satisfy [f, ξ] = 0 for all f ∈ OX , hence are OX -linearmaps OX → OX , that is, sections of OX .

For (ii), for any ξ ∈ F1DX , we claim that ξ′ = ξ − ξ(1) is a derivation OX → OX , hence a section ofTX . Indeed, for any two sections f0, f1 ∈ OX , we have

f1f0ξ(1)− f1ξ(f0)− f0ξ(f1) + ξ(f0f1) = 0

and therefore

ξ′(f0f1) = ξ(f0f1)− ξ(1)f0f1 = f1ξ(f0)− f0d(f1)− 2ξ(1)f0f1 = f0ξ′(f1) + f1ξ

′(f0).

The map F1DX → OX ⊕ TX given by ξ 7→ (ξ(1), ξ − ξ(1)) is then inverse to the evident addition mapOX ⊕ TX → F1DX , providing the desired isomorphism.

Proposition 2.5. The canonical map

SymOX TX → grDXis well-defined and an isomorphism.

Proof. For this, we must first check that grDX is commutative, for which it suffices to show that[Fn1DX , Fn2DX ] ⊂ Fn1+n2−1DX . Indeed, for d1 ∈ Fn1DX , d2 ∈ Fn2DX , and f0, . . . , fn1+n2−1 ∈ OX ,repeatedly applying the Jacobi identity gives that

[f0, [f1, · · · [fn1+n2−1, [d1, d2]]]]

is the sum of monomials of the form

[[fi1 , [fi2 , . . . , [fik , d1]]], [fj1 , [fj2 , . . . , [fjn1+n2−k, d2]]]],

where similarly to before i1, . . . , ik and j1, . . . , jn1+n2−k form a partition of 0, . . . , n1 + n2 + 1. Inthis case, if k > n1, then [fi1 , [fi2 , . . . , [fik , d1]]] = 0 and if k < n1, then [fj1 , [fj2 , . . . , [fjn1+n2−k

, d2]]] = 0.

On the other hand, if k = n1, then both [fi1 , [fi2 , . . . , [fik , d1]]] and [fj1 , [fj2 , . . . , [fjn1+n2−k, d2]]] are

differential operators of order 0, hence they commute. Thus in all cases, the monomial vanishes, so theentire sum vanishes and we see that [d1, d2] ∈ Fn1+n2−1DX , as desired.

2Although this may seem to allow for the possibility of circular logic, our proof of Proposition 2.5 restricts to the affine

case without reference to this lemma.


The decomposition of Proposition 2.4(ii) then yields a natural map of commutative OX -algebrasψ : SymOX TX → grDX . To show that it is an isomorphism, we construct an inverse map φi : griDX →Symi

OX TX inductively as follows. On gr0DX , the map φ0 is just the inverse of the isomorphism of

Proposition 2.4(i). Suppose now that we have defined maps φj for j < i that are isomorphisms on thejth graded piece. Then, consider the map

OX ⊗k FiDX → gri−1Dgiven by f ⊗ d 7→ [d, f ]. This map kills OX ⊗ Fi−1DX , hence defines a map

OX ⊗ griDX → gri−1DXφi−1

→ Symi−1OX TX

which for each element of griDX yields a derivation OX → gri−1DX . Therefore, we may take φi to bethe induced composition

φi : griDX → Derk(OX ,Symi−1OX TX) ' Symi−1

OX TX ⊗ TX → SymiOX TX .

That φi ψi = id is obvious, so to check that φi is inverse to ψi, we note that the map

griDX → gri−1DX ⊗ TXsends d ∈ griDX to some d′ ⊗ ξ ∈ TX ⊗ gri−1DX for which ξ(f)d′ = [d, f ]. This means then thatd′(ξ(f)) = ξ(f)d′(1) = d(f)− fd(1), so d′ξ = d− d(1), thus d′ξ = d in griDX , completing the check.

Proposition 2.6. The sheaf of OX-algebras DX is generated over OX by TX subject to the obviouscommutation relation [ξ, f ] = ξ(f) for sections ξ ∈ TX and f ∈ OX .

Proof. This is essentially a formal consequence of Proposition 2.5; we refer the reader to [Gai05, Propo-sition 5.3] for the complete proof.

Recall now that we are working over a variety X which we assumed to be smooth, which means thatwe may locally trivialize the tangent bundle TX of X. As a result, we recall that, locally on X, thereexist sections f1, . . . , fn ∈ OX such that their images dfi in T ∗X form a free basis for T ∗X ; we call such asystem an etale coordinate system. This technique will be quite useful for us in the future, particularlybecause it allows us to perform explicit calculations using the presentation of Proposition 2.6.

Proposition 2.7. Suppose that f1, . . . , fn ∈ OX form an etale coordinate system for X. Then, locallythere exist vector fields ξ1, . . . , ξn on X such that the sheaf of differential operators has the presentation

DX ' OX [ξ1, . . . , ξn]/〈[ξi, ξj ] = 0, [ξi, fj ] = δij〉.

Proof. It suffices to take ξi ∈ TX to be sections forming a dual basis to the basis dfi for T ∗X . That DXhas the desired presentation follows from Proposition 2.6.

We are now ready to define D-modules, the fundamental geometric objects of study in this section.

Definition 2.8. A (left, resp. right) DX-module on X is a quasi-coherent OX -module F equipped with a(left, resp. right) DX -action compatible with the OX -action on F. Denote by DMod(X) and DMod(X)r

the categories of left and right DX -modules.

Proposition 2.9. Let F be a quasi-coherent OX-module. Then, to give a left DX-module structure onF is to give a k-linear action of TX on F such that for sections f ∈ OX , ξ1, ξ2 ∈ TX , and m ∈ F, wehave

• (fξ) ·m = f(ξ ·m),• ξ · (fm) = ξ(f) ·m+ f(ξ ·m), and• [ξ1, ξ2] ·m = ξ1 · (ξ2 ·m)− ξ2 · (ξ1 ·m).

To give a right DX-module structure on F is to give a k-linear action of TX on F such that for sectionsf ∈ OX , ξ1, ξ2 ∈ TX , and m ∈ F, we have

• m(fξ) = f(mξ),• (fm)ξ = ξ(f) ·m− f(mξ), and• m[ξ1, ξ2] = mξ1ξ2 −mξ2ξ1.

Proof. This follows immediately from the presentation of DX in Proposition 2.6.

Using Proposition 2.9, we may now see some basic examples of D-modules.

Example 2.10. We may equip DX itself with both a left and right DX -module structure by left and rightmultiplication, as DX is quasi-coherent with respect to both the left and right OX -module structures byLemma 2.2.


Example 2.11. The structure sheaf OX is a DX -module with respect to the action of TX by derivations.

Example 2.12. This example provides some analytic motivation for the definition of a D-module. Forj = 1, . . . , k and i = 1, . . . , n, choose differential operators Dij ∈ DX and consider the system

(1)

n∑i=1

Dijfi = 0

of partial differential equations in the functions f1, . . . , fn. Construct a DX -module

M = coker(DnX

Dij→ DkX)

corresponding to this system. Then, we see that (global) solutions to (1) with fi ∈ OX correspondto morphisms of DX -modules M → OX . Further, any space of functions on X on which differentialoperators DX act corresponds exactly to a DX -module F; then (global) solutions to (1) in this space offunctions are simply morphisms of DX -modules M→ F.

In general, we will show later that any DX -module M which is coherent as an OX -module will belocally free. In this case, the conditions for M to be a D-module given in Proposition 2.9 translate exactlyinto the conditions for M to be endowed with a flat connection ∇ : TX → EndkM, where we define ∇simply to be the action of TX given by the DX -module structure.

Example 2.13. For any closed point x ∈ X, define the right DX -module δx to be generated by a formalelement 1x subject to the single relation

1x · f = 1x · f(x)

for f ∈ OX . By this, we mean that δx is isomorphic to

(ix)∗kx ⊗OXDX

as an OX -module, where the right DX -action is by right multiplication. Here the nomenclature δxreflects the fact that elements of OX act on 1x by evaluation at x. In particular, in the case whereX = A1 = Spec(k[t]) and x = 0 ∈ X, then we see that

δ0 ' 10 · k[∂t],

where the action of ∂t is by multiplication and the action of t is by ∂nt · t = n∂n−1t . In particular, we

note that the action of t annihilates δ0 as a right OX -module. We will see a similar construction to thisone later when we prove Kashiwara’s Theorem (Theorem 2.19).

2.2. Left and right D-modules. The goal of this subsection is to introduce an operation which willallow us to convert between left and right D-modules. We will see that this operation gives an equivalencebetween the categories of left and right DX -modules on a smooth variety X. Therefore, we will focusour attention on left D-modules. In particular, in the sequel, by a “D-module” we will always mean a“left D-module” unless we specify that it is a right D-module.

To give this construction, we must use some basic Hom and tensor operations on D-modules. Let usfirst see that these make sense.

Proposition 2.14. Let F,F′ be left DX-modules and G,G′ be right DX-modules. Then, the followingOX-modules acquire a left DX-module structure via the indicated action of TX :

• F ⊗OX F′ with action given by

ξ · (m⊗m′) = ξ(m)⊗m′ +m⊗ ξ(m′);• HomOX (F,F′) with action given by

(ξ · φ)(m) = ξ(φ(m))− φ(ξ(m));

• HomOX (G,G′) with action given by

(ξ · φ)(n) = −φ(n)ξ + φ(nξ).

The following OX-modules acquire a right DX-module structure via the indicated action of TX :

• G⊗OX F with action given by

(n⊗m)ξ = nξ ⊗m− n⊗ ξ(m);

• HomOX (F,G) with action given by

(φξ)(m) = φ(m)ξ + φ(ξ(m)).


Here ξ ∈ TX ,m ∈ F,m′ ∈ F′, n ∈ G, n′ ∈ G′ are sections of the corresponding sheaves.

Proof. In each case, we must check that the indicated action is well-defined and satisfies the relevantconditions of Proposition 2.9. We do this in the first case, leaving the rest of the verifications to thereader. Take a section f ∈ OX and ξ1, ξ2 ∈ TX . First, to show that the action is well-defined, it sufficesto note that

ξ · (fm⊗m′) = ξ(fm)⊗m′ + fm⊗ ξ(m′) = ξ(f)m⊗m′ + fξ(m)⊗m′ + fm⊗ ξ(m′)= fξ(m)⊗m′ +m⊗ ξ(fm′) = ξ · (m⊗ fm′).

We now check each condition in Proposition 2.9 in turn. The first is evident, the second follows from

ξ·(f ·m⊗m′) = ξ(fm)⊗m′+fm⊗ξ(m′) = ξ(f)m⊗m′+fξ(m)⊗m′+fm⊗ξ(m′) = ξ(f)m⊗m′+fξ(m⊗m′),

and the third from

[ξ1, ξ2] · (m1 ⊗m2) = [ξ1, ξ2]m1 ⊗m2 +m1 ⊗ [ξ1, ξ2]m2

= (ξ1ξ2 − ξ2ξ1)m1 ⊗m2 +m1 ⊗ (ξ1ξ2 − ξ2ξ1)m2

= (ξ1ξ2m1 ⊗m2 + ξ1m1 ⊗ ξ2m2 + ξ2m1 ⊗ ξ1m2 +m1 ⊗ ξ1ξ2m2)

− (ξ2ξ1m1 ⊗m2 + ξ1m1 ⊗ ξ2m2 + ξ2m1 ⊗ ξ1m2 +m1 ⊗ ξ2ξ1m2)

= (ξ1ξ2 − ξ2ξ1) · (m1 ⊗m2).

We may now give the correspondence between left and right D-modules. Let ΩX := det(T ∗X) denotethe sheaf of top dimensional differentials on X. Recall the natural action of TX on ΩX via the Liederivative Lieξ; it is given explicitly by

Lieξ(ω) :=(ξ1 ∧ · · · ∧ ξn 7→ ξ(ω(ξ1 ∧ · · · ∧ ξn))−

n∑i=1

ω(ξ1, . . . , ξi−1, [ξ, ξi−1], ξi+1, . . . , ξn)).

Lemma 2.15. The action ω · ξ = −Lieξ(ω) gives ΩX the structure of a right DX-module.

Proof. We must check the conditions of Proposition 2.9 using the explicit form of Lieξ. Having performedone such check in the proof of Proposition 2.14, we leave the gory details to the reader.

Consider now the OX -module

ΩDX := ΩX ⊗OXDX .

Note that ΩDX is endowed with two commuting structures of right DX -module via right multiplicationby DX and the construction of Lemma 2.14. Therefore, for any left DX -module F, the OX -module

ΩDX ⊗DXF ' ΩX ⊗

OXF

acquires a right DX -module structure in two equivalent ways, either by applying Proposition 2.14 tothe second expression or by considering the right DX -module structure on ΩDX in the first expression.These two descriptions are equivalent, giving a functor

ΩDX ⊗DX − ' ΩX ⊗OX −

from left DX -modules to right DX -modules. Similarly, we obtain a functor

HomDrX (ΩDX ,−) ' HomOX (ΩX ,−)

from rightDX -modules to leftDX -modules, where we note that for a rightDX -module G,HomDrX (ΩDX ,G)has two commuting structures of left DX -module coming from the two commuting structures of rightDX -module on ΩDX . These two functors will provide the claimed equivalence between left and rightDX -modules.

Proposition 2.16. The functors ΩDX ⊗DX − and HomDrX (ΩDX ,−) define an equivalence between thecategories of left and right DX-modules.

Proof. We first check that the functors are adjoint. For this, we claim that the standard Hom-tensoradjunction map for OX -modules restricts to a map

HomDrX (ΩX ⊗OX−,−)→ HomDX (−,HomOX (ΩX ,−)).


For this, it suffices to check that the adjunction map Ψ(φ) =(m 7→ (ω 7→ φ(ω ⊗m))

)restricts to an

isomorphism between maps of right DX -modules and maps of left DX -modules. Notice that

ξ ·Ψ(φ)(m) = ξ ·(ω 7→ φ(ω ⊗m)

)=(ω 7→ −φ(ω ⊗m)ξ + φ(ωξ ⊗m)

)and

Ψ(φ)(ξm) =(ω 7→ φ(ω ⊗ ξm)

),

thus Ψ(φ) is a map of left DX -modules if and only if

φ(ω ⊗m)ξ = φ(ωξ ⊗m)− φ(ω ⊗ ξm) = φ((ω ⊗m)ξ)

for all ω⊗m ∈ ΩX ⊗OX F. This occurs if and only if φ is a map of right DX -modules, giving the desiredadjunction.

It now remains to check that the unit and co-unit maps

ΩX ⊗OXHomOX (ΩX ,−) ' ΩDX ⊗DX

HomDrX (ΩDX ,−)→ id

and

id→ HomDrX (ΩDX ,ΩDX ⊗DX−) ' HomOX (ΩX ,ΩX ⊗

OX−)

are isomorphisms. On the level of OX -modules, this follows from the statement that HomOX (ΩX ,OX)is the dual line bundle to ΩX , thus it holds on the level of DX -modules as well.

In general, we find the D-module actions given by Proposition 2.14 on tensor products to be a greatdeal more intuitive than the ones onHom’s. Therefore, let us reformulate the functorHomDrX (ΩDX ,−) in

these terms. Define DΩX := HomDrX (ΩDX ,DX), which carries two commuting left DX -module structures,

one via the left action on DX and one via Proposition 2.14 as HomOX (ΩX ,DX). Then, for a right DX -module G, we see that

HomDrX (ΩDX ,G) ' G ⊗DXHomDX (ΩX ,DX) ' G ⊗

DXDΩX ,

where the left DX -module structure on G⊗DX DΩX is given by the left action on DX .3

2.3. Pullback and pushforward of D-modules. Consider now a smooth map φ : X → Y of smoothalgebraic varieties. In this subsection, we describe a way to pullback and pushforward D-modules alongφ. We will first define “standard” functors for the pullback and pushfoward in this subsection. However,these functors will not be well-behaved, and the “correct” versions which we define later will exist onlyin the derived category. A complication, however, is that only the derived pullback coincides with thederived functor of the standard pullback; the derived pushforward will be a closely related but differentfunctor, as the standard pushforward will be neither left nor right exact in general.

We now discuss some notational issues arising from this. We use φ∗ and φ∗ to denote the standardpushforward and pullback in the category of quasi-coherent O-modules and φ. and φ. to denote thepushforward and pullback on the level of sheaves. We denote the standard pullback of D-modules by φ∆

and the standard pushforward by φ+.4 Finally, we denote the derived pullback by φ! and the derivedpushforward by φ?.

3It is tempting here to consider the dual line bundle Ω−1X := HomOX (ΩX ,OX) to ΩX and use the isomorphism

HomOX (ΩX ,F) ' HomOX (ΩX ,OX) ⊗OX

F ' Ω−1X ⊗OX

F

of OX -modules to express HomDrX

(ΩDX ,−) as a tensor product over OX . However, we caution the reader that there is in

general no DX -module structure on Ω−1X . Indeed, for a smooth algebraic curve, this is a consequence of what is known as

Oda’s rule, which states that a line bundle on a curve of genus g has a left D-module structure if and only if it has degree0 and a right D-module structure if and only if it has degree 2g − 2. We refer the reader to [HTT08, Remark 1.2.10] for

more details.4We will see that φ∆ coincides with φ∗ on the level of O-modules. However, it is important to note that φ+ and φ?

will generally be different.


2.3.1. The definitions. First, let F be aDY -module. Define itsD-module pullback φ∆(F) to be isomorphicto φ∗(F) as an OX -module with DX -action induced by the map dφ : TX → φ∗TY corresponding to φ.Explicitly, this means that

φ∆(F) ' OX ⊗φ.(OY )

φ.(F)

with action of DX given according to the Leibnitz rule by ξ · (f ⊗m) = ξ(f)⊗m+f ·dφ(ξ)(m). Definingthe (DX , φ.(DY ))-bimodule

DX→Y := φ∆(DY ) = OX ⊗φ.(OY )

φ.(DY ),

we may express the D-module pullback of F in the form

φ∗(F) ' DX→Y ⊗φ.(DY )

φ.(F),

where the DX -action on φ∆(F) is simply the left DX -action on DX→Y .

Lemma 2.17. Let φ : Y → Z, ψ : X → Y be morphisms of smooth varieties. The D-module pullbacksatisfies the composition rule

ψ∆ φ∆ = (φ ψ)∆.

Proof. It suffices simply to check that

DX→Z ' DX→Y ⊗φ.DY

φ.DY→Z .

This is clear on the level of O-modules, so it remains only to notice that the D-module structure respectsthe O-module isomorphism.

Let us now try to imitate this construction to produce a D-module pushforward. For a right DX -module G, notice that φ.(G⊗DXDX→Y ) is naturally a right DY -module as the sheaf-theoretic pushforwardof a sheaf of φ.(DY ) modules. Apply the functors of Proposition 2.16 to translate this to a functor betweenleft D-modules, and call the result φ+. We see then that for a left DX -module F, we have

φ+(F) : = HomDY (ΩDY , φ.((ΩDX ⊗DXF) ⊗DXDX→Y ))

' φ.((ΩDX ⊗DXDX→Y ) ⊗

DXF) ⊗DYDΩY

' φ.((ΩDX ⊗DXDX→Y ⊗

φ.(DY )φ.DΩ

Y ) ⊗DX

F).

Therefore, defining the (φ.(DY ),DX)-bimodule

DY←X := ΩDX ⊗DXDX→Y ⊗

φ.(DY )φ.DΩ

Y ,

we obtain the D-module pushforward

φ+(F) = φ.(DY←X ⊗DX

F).

Remark. In the manipulations we did to reduce φ+(F) above, we made several moves which involvedpermuting the order in which we took tensor products involving ΩDX and DΩ

Y . We justify these forΩDX as follows; the argument for DΩ

Y will be entirely analogous. Note that the map τ : ΩDX → ΩDXextending

τ(ω ⊗ η) = −Lieηω ⊗ 1− ω ⊗ ηis an involution intertwining the two right DX -actions on ΩDX . Thus, any left DX -module F, τ inducesan isomorphism of OX -modules

ΩDX ⊗DXF ' Ω′DX ⊗DX

F,

where we denote by ΩDX and Ω′DX the object ΩDX with its two different right DX -actions. For instance,in the manipulation above, the isomorphism

(ΩDX ⊗DXF) ⊗DXDX→Y ' (ΩDX ⊗DX

DX→Y ) ⊗DX

F

simply exchanges the choice of which right DX -action on ΩDX each of F and DX→Y are tensored over.

In the definition of the D-module pushforward, observe that φ+(F) is the composition of the left exactfunctor φ. and the right exact functor DY←X ⊗DX −. As a result, φ+ is not well-behaved in general. Forinstance, a composition rule analogous to Lemma 2.17 does not hold . However, when φ is affine, φ. isexact, and this pushforward is better behaved. We now examine a particular instance of this in detail.


2.3.2. Closed embeddings and Kashiwara’s theorem. Let us now suppose that φ : X → Y is a closedembedding. In particular, this means that φ is affine, hence φ. is exact and φ+ is right exact. In thiscase, we may define a different pullback functor

φ+(F) := Homφ.DY (DY←X , φ.F),

where the left DX -module structure on φ+(F) comes from the right DX -module structure on DY←X .We will later see that φ+ is related to the standard pullback functor in the derived category. For now,let us first relate it to φ+.

Proposition 2.18. If φ : X → Y is a closed embedding, we have an adjoint pair of functors

φ+ : DMod(X) DMod(Y ) : φ+.

Proof. The chain of natural isomorphisms

HomDY (φ.(DY←X ⊗DX−),−) ' Homφ.DY (DY←X ⊗

DX−, φ.−) ' HomDX (−,Homφ.DY (DY←X ,−))

gives the result, where the first follows from the fact that φ.(F) is supported on X for all DY -modules F,hence applying φ. is an isomorphism on Hom’s out of φ.(F) and the second follows from (taking globalsections of) the Hom-tensor adjunction

Homφ.DY (DY←X ⊗DX−, φ.−) ' HomDX (−,Homφ.DY (DY←X ,−)).

As we saw in the proof of Proposition 2.18, any DY -module in the image of φ+ is supported on X ⊂ Y .Denote by the DMod(Y )X the full subcategory of DMod(Y ) generated by the DY -modules with supportlying in X. The following central result in the theory of D-modules shows that φ+ defines an equivalenceof categories onto DMod(Y )X .5

Theorem 2.19 (Kashiwara’s Theorem). If φ : X → Y is a closed embedding, we have an equivalenceof categories

φ+ : DMod(X) DMod(Y )X : φ+.

Proof. Having shown that φ+ and φ+ form an adjoint pair, it suffices to check that the unit and counitare isomorphic to the identity. This is a local assertion, so we reduce to the case where X = Spec(A/I)and Y = Spec(A). Because X and Y are smooth varieties, we can find a regular sequence I = (f1, . . . , fn)generating the defining ideal I of X. To prove the assertion for X → Y , it suffices to prove it for eachembedding in the chain

X → X1 → X2 → · · · → Xn−1 → Y,

where Xi = Spec(A/(fi, . . . , fn)).We have therefore reduced to the case where X = Spec(A/f) and Y = Spec(A) for some regular

element f ∈ A. Because A is regular, after further localization we may find an etale coordinate systemf1, . . . , fn where X is cut out locally by the vanishing of the last coordinate (by extending the equationdefining X to a regular sequence in A). In this case, we see that

DX = OX [ξ1, . . . , ξn−1]/〈[ξi, fj ] = δij〉and

DY = OY [ξ1, . . . , ξn−1, ξ]/〈[ξi, fj ] = δij〉,where OX = OY /(f) for f = fn and ξ = ξn. Let us now compute DY←X in these coordinates, keepingtrack of the D-module structures. Notice that

DX→Y = OX ⊗φ.OY

φ.DY ' OX [ξ1, . . . , ξ]/〈[ξi, fj ] = δij〉 ' DX [ξ]

with the left DX and right φ.DY -actions given by left and right multiplication in the obvious way. Now,we see that

DY←X = ΩDX ⊗DXDX→Y ⊗

φ.DYφ.DΩ

Y ' OX [ξ1, . . . , ξn]/〈[ξi, fj ] = δij〉 = [ξ]DX

with the left φ.DY and right DX actions given again by left and right multiplication. We recall here thatξ was defined to satisfy [ξ, f ] = 1.

5This is quite different from the situation for O-modules. If X ⊂ Y is defined by the sheaf of ideals IX ⊂ OY , it is

possible that a OY -module F supported on X only vanishes upon repeated multiplication by IX , hence does not lie inthe image of the O-module pushfoward. For a simple example, take Y = Spec(A) affine and X = Spec(A/f) defined by a

single equation; then F := A/(fn) is supported at X, but is not a pushforward of any quasicoherent OX -module.


Having set up our coordinates, we may proceed to checking the isomorphisms. For any DX -moduleF, the map F → φ+φ+F is given by

F → Homφ.DY ([ξ]DX , φ.φ.([ξ]DX ⊗DX

F)) ' Homφ.DY ([ξ]DX , [ξ]F).

Because φ.DY acts surjectively on [ξ]DX , a section of the right hand side is specified by a sections =

∑i ξimi of [ξ]F killed by f . But because f ·m = 0, we see that

f ·∑i

ξimi =∑i

iξi−1mi 6= 0

unless mi = 0 for i > 0. This means that s must lie in the grade 0 component F of [ξ]F and hence thatthe map F → φ+φ+F is an isomorphism.

It remains then to check that for a DY -module F supported on X, the evaluation map

φ+φ+(F) ' φ.([ξ]DX ⊗

DXHomφ.DY ([ξ]DX , φ.F))→ F

is an isomorphism. For this, we must analyze the structure of F more carefully.

Consider the operator T : Ffξ→ F given by the action of fξ and define Fk := ker(F

T−k→ F). We seeimmediately that for m ∈ Fk, we have

fξ · fm = (f2ξ + f)m = (k + 1)fm

and

fξ · ξm = (ξfξ − ξ)m = (k − 1)ξm,

meaning that fm ∈ Fk+1 and ξm ∈ Fk−1. Thus, we may consider the diagram

(2) · · ·ξ -f

F0ξ -f

F−1ξ -f

F−2ξ -f

F−3ξ -f

· · ·

where we have ξf − fξ = 1 as maps Fk → Fk for each k. By definition fξ acts by k on Fk, so

ξf acts by k + 1 on Fk. In particular, this implies that for k ≤ −1, the map Fkξ→ Fk−1 is an

isomorphism, and for k ≤ −2, the map Fk → Fk+1 is an isomorphism. We may therefore conclude thatF−1 ' F−2 ' F−3 ' · · · .

It remains now to check that F ⊂ F−1⊕F−2⊕· · · . Because F is supported on X, we have a filtration

F =

∞⋃k=1

Fk,

where Fk := ker(Ffk→ F). Note that F1 is non-empty because the map F

f→ F is not injective. Wewill now induct on k to show that Fk ⊂ F−1 ⊕ · · · ⊕ F−k. For the base case k = 1, if m ∈ F1, thenTm = fξm = ξfm−m = −m, hence F1 ⊂ F−1. Suppose now that Fk−1 ⊂ F−1 ⊕ · · · ⊕ F−k+1 for somek and take m ∈ Fk. Then, notice that fm ∈ Fk−1 ⊂ F−1 ⊕ · · · ⊕ F−k+1, hence ξfm ∈ F−1 ⊕ · · · ⊕ F−k.Similarly, we see that ξf2ξm = ξfξfm−ξfm ∈ F−1⊕· · ·⊕F−k because fm ∈ Fk−1, so by the inductionhypothesis and (2), we see that fξm ∈ F−1 ⊕ · · · ⊕ F−k, meaning that

m = ξfm− fξm ∈ F−1 ⊕ · · · ⊕ F−k.

Thus, we have obtained an isomorphism of DY -modules

(3) F =

∞⊕k=1

F−k ' [ξ]F−1,

where f acts by 0 on F−1.6

6We note here the evident formal similarity between F ' [ξ]F−1 and the D-module δx of Example 2.13. Indeed, triviallygeneralizing the latter construction to apply to any closed embedding i : Y → X, we obtain for any DY -module M a right

DX -module i∗M ⊗OX DX (which is the pushforward of right D-modules). In the situation of the present proof, if we

take M = F−1 and then apply the equivalence between left and right DX -modules, the result is exactly the DX -moduleF, which shows that any DX -module supported on some closed smooth subvariety Y cut out by a single equation may be

realized as the pushforward of a DY -module. This is the key idea behind the proof we are giving of Kashiwara’s Theorem.


Returning to our original objective, we see that

φ+φ+(F) ' φ.([ξ]DX ⊗

DXHomφ.DY ([ξ]DX , φ.F))

' φ.([ξ]DX ⊗DXHomφ.DY ([ξ]DX , φ.[ξ]F−1))

' φ.([ξ]DX ⊗DX

φ.F−1),

where the final isomorphism follows because a section of Homφ.DY ([ξ]DX , φ.[ξ]F−1)) is specified by asection of [ξ]F−1 killed by f , which must lie in F−1 itself. Under this identification, the map φ+φ

+(F)→ F

is simply multiplication, thus an isomorphism by (3).

Remark. We have thus far only considered D-modules on smooth algebraic varieties. While our defini-tion of the sheaf of differential operators DX did not depend on the fact that X was smooth, a similardefinition on singular varieties is pathological. Instead, we sketch an approach suggested by Kashiwara’sTheorem. We may locally realize any singular variety X as a closed subvariety X → Y of some smoothsubvariety Y . Then, we define the category DMod(X) := DMod(Y )X locally to be the category ofDY -modules supported on X. To check that our definition did not depend on the choice of Y , for twosuch embeddings X → Y1 and X → Y2, take a third smooth variety Z such that the two embeddings fitinto the commutative diagram

X - Y1

Y2

?- Z.?

Then, we have equivalences of categories

DMod(Y1)X ' (DMod(Z)Y1)X ' DMod(Z)X ' (DMod(Z)Y2

)X ' DMod(Y2)X

by Kashiwara’s theorem on the embeddings of smooth varieties Y1 → Z and Y2 → Z. It remains to gluethese local constructions we have made into a global category on all of X and to check that the gluingprocedure is independent of any choices we made. This will follow from some further abstract nonsensewhich we suppress. We do note here, however, that the category DMod(X) we have just defined is apriori unrelated to the category of DX -modules when X is singular, and there may not be functors ineither direction.

Kashiwara’s Theorem suggests that the behavior of D-modules is much more strongly structured thanthe behavior of OX -modules. We illustrate this with the following immediate consequence.

Proposition 2.20. If a DX-module F is coherent as an OX-module, then it is locally free as an OX-module.

Proof. It suffices to show that F is flat, for which it suffices to check that the dimensions of the fibersF/mxF of F are locally constant. We claim it suffices to check this for the restriction of F to any non-singular curve i : C → X embedded into X; such a restriction will be OX -coherent because i∆ and i∗

coincide in this case (as the set of points which may be connected to a given point by a smooth curve isopen).

We have now reduced to the case where X is a curve. We claim that F is torsion-free on X; indeed,if F has torsion at a point x, then this means there is a non-zero G ⊂ F such that mNx kills G. That is, Gis supported on x, so by Kashiwara’s Theorem we may write

G = (ix)+(H) = (ix).(DX←x ⊗kH)

for some Dx-module H (which is nothing more than a vector space). But recall from the proof ofKashiwara’s Theorem that DX←x ' k[ξ] as a vector space over k, hence G = (ix).(H[ξ]) is not coherentas an OX -module, a contradiction.

Because X was a curve, each stalk Fx is a module over OX,x, which is a regular local ring of dimension1, hence a DVR. But a module over a DVR is free if and only if it is torsion free, meaning that Fx is freeover OX,x for each x. We conclude that F is locally free on X, so the dimensions of its fibers F/mxF onX are locally constant, as needed.

Remark. We distinguish here between the notions of a coherent DX -module, which has not yet beendefined and will appear in Subsection 2.4, and a DX -module which is coherent as an OX -module.


2.3.3. The derived definitions. Let us now return to our original goal of defining the pullback and push-forward of D-modules. As we said before, these operations should take place in the (bounded) derivedcategory D(DMod(X)) of DX -modules. For this to make sense, we must make an additional assumptionthat all varieties we consider are quasi-projective, meaning they admit a locally closed embedding intosome projective space. Then, we have the following technical result.

Proposition 2.21. If X is quasi-projective, the category DMod(X) has enough injectives and localprojectives and has finite homological dimension.

Proof. This is a combination of [HTT08, Proposition 1.4.14] and [HTT08, Corollary 1.4.20].

By Proposition 2.21, we may expect the operations of derived Hom and tensor to be well-behaved.Let D(DMod(X)) := D(DMod(X)) be the bounded derived category of DMod(X). Then, we may define

RHomDX (−,−) and −L⊗DX − in the usual way. Now, let φ : X → Y be a morphism. We define the

derived pullback φ! as

φ!(−) := Lφ∆(−)[dimX − dimY ] = DX→YL⊗

φ.DYφ.(−)[dimX − dimY ]

and the derived pushforward φ? as

φ?(−) := Rφ.(DY←XL⊗DX−),

which will be well-defined as maps between the bounded derived categories. Let us see what thesedefinitions mean in some special cases.

Example 2.22. If φ is an open embedding, then φ∆ is exact, and

DX→Y ' OX ⊗φ.OY

φ.DY ' OX ⊗OXDX ' DX

by quasicoherence of DY . Thus, we see that φ! ' φ.[dimX − dimY ]. Similarly, note that

DY←X ' ΩDX ⊗DXDX→Y ⊗

φ.DYφ.DΩ

Y ' ΩDX ⊗DXDΩX ' DX ,

which implies that φ+ ' φ. ' φ∗ is the usual pushfoward and φ? ' Rφ∗.Example 2.23. If φ is a closed embedding, then taking locally a regular sequence in the sheaf of idealsdefining X, we can consider the Koszul resolution

0→ KdimY−dimX ⊗φ.OY

φ.DY → · · · → K0 ⊗φ.OY

φ.DY → DX→Y → 0,

which gives a locally free resolution for DX→Y as a right φ.DY -module.7 Thus, to compute φ!, we take

φ!(F) ' DX→YL⊗

φ.DYφ.F ' K• ⊗

φ.OYφ.DY

L⊗

φ.DYφ.F ' K• ⊗

φ.OYφ.F.

For φ?, we saw that φ+ was exact, and the proof of Kashiwara’s Theorem showed that DY←X is locallyfree as a right DX -module, hence we see that

φ?(−) ' Rφ.(DY←XL⊗DX−) ' Rφ.(DY←X

L⊗DX−) ' Rφ+ ' φ+.

Example 2.24. Suppose now that φ is the map X → pt. Then, we see that

DX→pt ' OX and Dpt←X ' ΩDX ⊗DXOX ' ΩX ,

where there is a resolution

(4) 0→ Ω0X ⊗OXDX → Ω1

X ⊗OXDX → · · · → ΩnX ⊗

OXDX → ΩX → 0

of locally free right DX -modules (see [HTT08, Lemma 1.5.27]), where Ω1X := T ∗X . The resolution (4) is

a form of the Spencer resolution. Therefore, we see that

φ?(−) ' Rφ.(ΩXL⊗DX−) ' Rφ.(Ω•X ⊗

OXDX ⊗

DX−) ' Rφ.(Ω•X ⊗

OX−).

Taking the special case of φ?(OX), we see that φ?(OX) ' Rφ.(Ω•X) and therefore that

RiΓ(φ?(OX)) ' Riφ.(Ω•X) ' Hi(X,Ω•X),

7We note here that this particular instance is given by tensoring the ordinary Koszul resolution from commutative

algebra by DY .


which is the algebraic de Rham cohomology of X (where Hi denotes the hypercohomology). In particular,when X is defined over C, then by Grothendieck’s algebraic de Rham theorem [Gro66, Theorem 1],RiΓ(φ?(OX)) gives Hi

DR(X,C), the de Rham cohomology of the associated complex algebraic variety.

2.4. The singular support. We now give a construction that allows us to consider a second notionof support for a class of DX -modules satisfying a certain local finiteness condition. The relevant notionarises from the observation of Proposition 2.5 that grDX ' SymOX TX ' OT∗X , suggesting that we maywish to view a DX -module in terms of a related graded module over SymOX TX , where we note thatSymOX TX ' OT∗X . We may then consider its support on the cotangent bundle T ∗X of X. Let us nowcarry out these constructions more rigorously.

2.4.1. The definition. Recall the order filtration F0DX ⊂ F1DX ⊂ · · · ⊂ DX on DX defined in Subsection2.1. For a DX -module F, we say that a filtration FiF on F is compatible with FiDX if FiDX ·FjF ⊂ Fi+jFfor all i, j. For any DX -module F equipped with a filtration, we see that grF F acquires the structureof a graded grDX ' OT∗X -module. We would like to define a notion of support for grF F which dependsonly on F and not on the choice of filtration; for this, we require the following key definition.

Definition 2.25. A filtration F0F ⊂ F1F ⊂ · · · of F is good if one of the following two equivalentconditions hold:

(i) grF F is a coherent OT∗X -module, or(ii) FiF is a coherent OX -module for all i and F1DX · FiF = Fi+1F for i large.

Lemma 2.26. The two characterizations of good filtrations given in Definition 2.25 are equivalent.

Proof. That (ii) implies (i) is obvious. To show (i) implies (ii), pick a set of generators [fi] for grF F

over OT∗X , and let fi ∈ F be a representative of each generator, where fi ∈ Fki − Fki−1 for a unique ki.Then, there is some N such that fi ∈ FNF for all i. Now, choose locally a set of generators ξj for TXover OX and for K ≥ N consider the set of elements

SK = ξj1 · · · ξjk · fi | k ≤ K − ki .

We claim that the obvious map O⊕SKX → FKF is surjective; indeed, filtering O⊕SKX by the maximalorder of a non-zero element corresponding to s ∈ SK , the map respects filtrations and is surjective onthe associated graded because [fi] generated grF F. This shows that FKF is coherent for all K ≥ N .Further, because SK+1 ⊂ F1DX ·SK , we obtain also that F1DX ·FiF = Fi+1F for i ≥ N , giving (ii).

Let now F be a DX -module equipped with a good filtration FiF. By definition, grF F is then acoherent OT∗X -module, so we may then define the singular support of F to be

SS(F) := supp(grF F),

the support of grF F. As is implicit in the definition, we must check that this definition is independentof the choice of good filtration on F.

Proposition 2.27. The singular support of F does not depend on the choice of good filtration on F.

Proof. Take two good filtrations F 1F and F 2F, and denote by SS1(F) and SS2(F) the (a priori different)singular supports associated to them. We now use an elegant trick which we believe is due to Bernstein.Say that the two filtrations are neighbors if

(5) F 1i ⊂ F 2

i ⊂ F 1i+1 ⊂ F 2

i+1

for all i. We first claim that if F 1 and F 2 are neighbors, then SS1(F) = SS2(F). Indeed, condition(5) means that the identity map F → F is compatible with the two filtrations, hence it induces a mapgr1 F → gr2 F. Let us place this map in an exact sequence

0→ K → gr1 F → gr2 F → C → 0,

where we notice that K =⊕

i F2i F/F

1i F and C =

⊕i F

2i+1/F

2i+1F. In particular, C and K are isomorphic

up to a shift in grading. Therefore, if x /∈ supp(gr1 F), then Kx = 0 and therefore Cx = 0, meaningthat (gr2 F)x ' (gr2 F)x = 0. Similarly, if x /∈ supp(gr2 F) then, we see also that x /∈ supp(gr1 F), whichimplies that SS1(F) = supp(gr1 F) = supp(gr2 F) = SS2(F), as desired.

It remains now to check that any two good filtrations F 1 and F 2 on F are linked up to a shift by a chainof neighboring filtrations (as the singular support is obviously invariant under shifts of the filtration).Indeed, define the filtrations

Gki F := F 1i F + F 2

i+kF


for all k ∈ Z. It is clear that Gk and Gk+1 are good and are neighbors for all k. Now, choose K sothat F1DX · F 1

i F = F 1i+1F and F1DX · F 2

i F = F 2i+1F for i ≥ K. Take N so that F 2

KF ⊂ F 1NF and

F 2KF ⊂ F 2

NF. Then, we claim that G−N = F 1 and that GN is a shift of F 1. For the first claim, ourchoice of K and N implies that F 2

i−NF ⊂ F 2i−N+KF ⊂ F 1

i F, we see that

G−Ni F = F 1i F + F 2

i−NF = F 1i F.

For the second claim, the choice of K and N gives F 1i F ⊂ F 1

i+KF ⊂ F 2i+NF, hence

GNi F = F 1i F + F 2

i+NF = F 2i+NF.

Thus, we see that F 1 and a shifted version of F 2 are linked by neighboring good filtrations, completingthe proof.

By Proposition 2.27, when a good filtration exists, we have a well-defined notion of singular support.We would like now to characterize when this happens. We say that a DX -module is coherent if it islocally finitely generated (over DX). It will turn out that these are precisely the DX -modules for whicha good filtration exists. We give some basic properties of coherent DX -modules below.

Lemma 2.28. Any coherent DX-module F on X is generated by a submodule which is coherent as anOX-module.

Proof. Recall that X was assumed to be quasi-projective, hence quasi-compact, so we may cover X bya finite collection of affines X =

⋃i Ui such that F|Ui is finitely generated as a DUi-module over Ui. On

each Ui, pick a set of generators for F over DUi and let Gi be the OUi-submodule generated by them.Then there exists a OX -submodule Fi of F such that Fi|Ui = Gi. Set F′ =

∑i Fi ⊂ F; notice that F′ is

coherent as the finite sum of coherent submodules of F. It is clear that F′ generates F as a DX -module,as needed.

Lemma 2.29. Let U ⊂ X be an open set and F a DX-module. Then, if F|U is coherent as a DU -module,there exists a coherent submodule F′ ⊂ F such that F′|U = F|U .

Proof. Take a OU -coherent submodule G ⊂ F|U that generates it as a DU -module. Then, we may find aOX -coherent submodule G′ of F such that G′|U ' G. Let F′ be the DX -submodule of F generated by G′;it is clearly coherent and satisfies F′|U ' F|U , as needed.

Using these, we can now show that good filtrations exist exactly for coherent DX -modules.

Proposition 2.30. A DX-module F admits a good filtration if and only if it is coherent.

Proof. If F admits a good filtration FiF, then picking K so that F1DX · FiF = Fi+1F for i ≥ K, a setof OX -generators for FKF will be a set of DX -generators for F, showing that F is coherent. Conversely,if F is coherent, take by Lemma 2.28 an OX -coherent submodule F′ ⊂ F that generates it and considerthe filtration

FiF := FiDX · F′.Because F1DX ·FiDX = Fi+1DX for all i by Proposition 2.5, we see that F1DX ·FiF = Fi+1F for all i.8

Now, because F0F = F′ is coherent, it follows that FiF = (F1DX)i · F0F is coherent for each i, showingthat FiF is a good filtration.

2.4.2. Bounding the singular support. Having now defined the singular support for coherent D-modules,let us characterize its behavior. In general, it is somewhat difficult to compute, but the following specialcase is simple and important.

Example 2.31. Suppose that F is OX -coherent. Then the trivial filtration on F with FiF = F is good,meaning that grF F ' F, where the action of TX is trivial. This implies that SS(F) is contained in theimage of the zero section X → T ∗X .

As we see from Example 2.31, DX -modules which are coherent as OX -modules have singular supportcontained in X. According to the following result of Bernstein, this is “smallest possible” in the followingsense.

Theorem 2.32 (Bernstein’s Inequality). For any non-zero coherent DX-module F, we have SS(F) ≥dim(X).

8In fact, Proposition 2.5 shows that FiDX is a good filtration.


We defer the proof of Theorem 2.32 for a moment to present and motivate a key result used in it.The basic idea of the proof will be to reduce by induction to the case where dim supp(F) = dim(X) byrestricting the base of F to (an open subset) of its support using Kashiwara’s Theorem. To accomplishthis, we need to understand how the singular support transforms under pullback and pushforward.Unfortunately, this is meaningless in general, as the (derived) pushforward and pullback do not alwayspreserve coherence. However, it holds in the following special cases, which will be enough for us to proveBernstein’s inequality.

Proposition 2.33. Let φ : X → Y be a morphism of varieties. Then, we have the following:

(i) if φ is a closed embedding, then a DX-module F is coherent if and only if φ+(F) is, and in thiscase we have

dimSS(φ+(F))− dim(Y ) = dimSS(F)− dim(X);

(ii) if φ is an open embedding, then for a coherent DY -module F, φ∆(F) is coherent, and in this casewe have

dimSS(φ∆(F)) = dimSS(F).

Proof. For (i), as in the proof of Kashiwara’s Theorem, we may reduce to the situation where Y is affinewith a etale coordinate system f1, . . . , fn with corresponding dual vector fields ξ1, . . . , ξn and X is definedby the vanishing of the last coordinate f = fn. As before, set ξ = ξn. We computed previously that inthis case φ+(F) ' [ξ]F, thus any good filtration on φ+(F) restricts to a good filtration on F and for agood filtration FiF, we may construct the good filtration

Fnφ+(F) :=∑i+j=n

ξi · FjF

on φ+(F) such that

grn φ+(F) =⊕i+j=n

ξi · grj F.

This shows that φ+(F) is coherent if and only if F is. To show that φ+ preserves dimSS(F)− dim(X),note that f kills grφ+(F) because f ·ξi = ξi−1f and f kills grF. Because the action of all other generatorsof SymOY TY commute with ξ, we find that

AnnY (grφ+(F))/(f) ' AnnX(F),

where f is regular in AnnY (grφ+(F)). This shows as needed that

dimSS(φ+F)− dim(Y ) = dim(Y )− dim AnnY (grφ+(F))

= dim(X) + 1− dim AnnX(F)− 1

= dimSS(F)− dim(X).

For (ii), because φ∆ coincides with the O-module pullback, it preserves coherence, as the pullbackof a good filtration will be a good filtration. Further, φ∆ induces an open embedding φ[ : T ∗X → T ∗Y ,hence by exactness of φ∆ we have an isomorphism gri φ∆F ' φ∗[ gri F. This shows that SS(φ∆F) 'SS(F) ∩ φ[(T ∗X), giving the claim.9

With Proposition 2.33 in hand, we are now ready to give the proof of Theorem 2.32.

Proof of Theorem 2.32. Given Kashiwara’s Theorem and Proposition 2.33, the proof is relatively simple.Proceed by induction on dim(X); if dim(X) = 0, the statement is vacuous. Now, for dim(X) > 0, supposethat the claim holds in smaller dimensions, and restrict to an open subset of X so that supp(F) consistsof a single irreducible component. By Proposition 2.33(ii), this preserves the hypotheses and desiredconclusions of the claim.

Now, if supp(F) = X, then we see that X ⊂ SS(F) (under the embedding of X → T ∗X via the zerosection), meaning that dimSS(F) ≥ dim(X). Otherwise, Z = supp(F) is a proper closed subset of Xof codimension at least 1. Pick some open set U of X so that U ∩ Z is non-empty and smooth, andwrite j : U → X and i : U ∩ U ∩ Z → U for the inclusions. Then, j∆F is supported on U ∩ Z, so byKashiwara’s Theorem we may find some DU∩Z-module G such that i+G = j∆F. By Proposition 2.33, wesee that j∆F and G are both coherent and thus that

dimSS(F)− dim(X) = dimSS(j∆F)− dim(U) = dimSS(G)− dim(U ∩ Z) ≥ 0

9We note that (ii) may be generalized much further (with appropriate modifications to the conclusion); in particular,we may consider any smooth morphism φ. We will only use the simpler case where φ is an open embedding, however, so

we restrict to it here.


by the induction hypothesis.

By Bernstein’s inequality, we see that DX -modules which are coherent as OX -modules must havesingular support SS(F) ⊂ X of dimension exactly X, meaning that their singular supports consist ofsome of the irreducible components of X. Decomposing the singular support into irreducible componentsin this manner is a fruitful general technique, as we demonstrate in the following proposition.

Proposition 2.34. Let G ⊂ F be coherent DX-modules. Then, we have

SS(F) = SS(G) ∪ SS(F/G),

and further this decomposition respects the multiplicities of the irreducible components on both sides.

Proof. A good filtration on F induces good filtrations on G and G/F by restriction and projection suchthat the short exact sequence

0→ G→ F → G/F → 0

is compatible with filtrations. Passing to the associated graded, we obtain a short exact sequence

0→ grG→ grF → grG/F → 0,

from which it follows that SS(F) = SS(G) ∪ SS(F/G).Let us now see a refinement of this statement. Observe that this decomposition respects the irreducible

components of SS(F). That is, for every irreducible component Ii of SS(F) with generic point pi, weobtain a short exact sequence

0→ (grG)pi → (grF)pi → (grG/F)pi → 0

of Artinian modules of finite length, where the length of (grF)pi is by definition the multiplicity mi(F)of Ii in grF. Then this short exact sequence implies that

mi(F) = mi(G) +mi(G/F)

for each irreducible component Ii of SS(F).10

Corollary 2.35. Any holonomic D-module F has finite length.

Proof. This is immediate from the result about irreducible components in Proposition 2.34, as the lengthof F is bounded by the sum of the multiplicities of the irreducible components of SS(F).

2.5. Holonomic D-modules and duality. In view of Bernstein’s Inequality from the previous section,it is natural to make the following definition of DX -modules which are of minimal size in some sense.

Definition 2.36. A coherent DX -module F is holonomic if and only if dimSS(F) = dim(X).

We shall see that the class of holonomic D-modules is particularly well-behaved. The following propo-sition shows that they are very close to O-coherent D-modules.

Proposition 2.37. Let F be a holonomic DX module. Then, there exists an open dense subset U ⊂ Xsuch that F|U is OU -coherent.

Proof. Set M := grF, and let M0 ⊂ M be the submodule of M supported away from X ⊂ T ∗X , wherewe interpret X as the zero section; that is, M0 is the submodule annihilated by the ideal of elementsof positive grade. Now, notice that T ∗X −X carries a Gm-action by multiplication on fibers and furtherthat the quotient Y of T ∗X −X under this action exists. Observe now that M0 is a graded module overSymOX TX supported on T ∗X −X; thus, it defines a sheaf on Y , and its support supp(M0) is invariant

under the Gm-action on fibers. This means that the fibers of supp(M0) have dimension at least 1 underthe projection π : supp(M0)−X → X, implying that

dim(X) ≥ dim supp(M) ≥ dim supp(M0) > dimπ(supp(M0)).

Then, letting U = X − π(supp(M0)), we see that U is open dense and that supp(M0) is disjoint fromπ−1(U), thus grF|U is supported on T ∗U ∩ (U ∪π−1(U)) ⊂ U . But this means that the good filtration onF|U is eventually constant, hence F|U is OU -coherent.

10Technically speaking, we should have showed that mi(F) was well-defined independent of the choice of good filtrationon F. To show this, the argument we used to prove Proposition 2.27 works verbatim with every instance of x /∈ supp(grF)

replaced by (grF)pi = 0.


Let us now see that holonomic D-modules behave well under the various operations on D-moduleswe have defined so far. By Proposition 2.34, it is clear that quotients and extensions of holonomic D-modules remain holonomic. For the rest of our compatibility properties, we must pass to the derivedcategory of holonomic D-modules, which we define on a variety X to be the subcategory D(DMod(X)hol)of D(DMod(X)) consisting of complexes whose cohomologies are holonomic. Similarly, we may definethe derive category of coherent D-modules, which we denote by D(DMod(X)c) The main compatibilityresult on holonomic D-modules is that this category is well behaved under the derived pushforward andpullback. Presenting a complete proof of Theorem 2.38 is beyond the ken of this essay, but we will givesome indication of the ingredients.

Theorem 2.38 ([HTT08, Theorem 3.2.3]). Let φ : X → Y be a morphism of varieties. Then, φ! mapsD(DMod(Y )hol) to D(DMod(X)hol) and φ? maps D(DMod(X)hol) to D(DMod(Y )hol).

2.5.1. The duality functor. We must first take a detour toward a duality functor which is definedfor all coherent D-modules, which is itself of independent interest. Define the duality functor DX :D(DMod(X)c)r → D(DMod(X)c) by

DX(F•) := RHomDX (F•,DΩX)[dimX].

To compute DX , we may take a locally projective resolution P• → F•, for which we have

DX(F•) = DX(P•) = HomDX (P•,DΩX)[dimX].

Therefore, we see that

D2X(F•) = D2

X(P•) ' HomDX (HomDX (P•,DΩX)[dimX],DΩ

X)[dimX]

' HomDX (HomDX (P•,DΩX),DΩ

X),

is locally isomorphic to P• under the map P• → HomDX (HomDX (P•,DΩX),DΩ

X) given by evaluation(because P• was locally projective and DΩ

X locally isomorphic to DX). The duality functor will turn outto have particularly nice behavior on holonomic D-modules, as we summarize below.

Theorem 2.39. If F is a coherent DX-module, then

(i) F is holonomic if and only if DX(F) lies in grade 0, and(ii) DX provides a duality of categories between DMod(X)hol and DMod(X)hol,r.

The proof of Theorem 2.39 relies crucially on the homological result attributed to J. E. Roos, whichis proved by passing between resolutions of F and grF F for good filtrations on F.

Theorem 2.40 ([HTT08, Theorem 2.6.7]). Let F be a coherent DX-module. Then, we have

(i) codimSS(Ext iDX (F,DΩX)) ≥ i, and

(ii) Ext iDX (F,DΩX) = 0 for i < codimSS(F),

where we recall that Ext iDX (F,G) is the cohomology in grade i of RHomDX (F,G).

We omit the proof but now show how it implies Theorem 2.39. In fact, Theorem 2.40 has quitepowerful consequences for duality in the derived category of all coherent DX -modules, but we wish tofocus on the holonomic case for now.

Proof of Theorem 2.39. This will be simple given the characterization given by Theorem 2.40. To wit,for (i), first suppose F is holonomic. Then, by Theorem 2.40(ii), we see that Hi(DX(F)) = 0 for i < 0(note the shift in grade due to to shift in the definition of DX). Further, by Theorem 2.40(i), we havethat

dimSS(Ext iDX (F,DΩX))− dim(X) ≤ dim(X)− i

for all i, which shows by Bernstein’s inequality that

Ext iDX (F,DΩX) = Hi−dimX(D(X)) = 0 for i > dim(X),

hence Hi(DX(F)) = 0 for i > 0 and DX(F) lies only in grade 0.On the other hand, if DX(F) lies only in grade 0, then by duality we see that

F ' DX(DX(F)) ' DX(DX(F)0),

so applying Theorem 2.40 to DX(F), we see that

2 dim(X)− dimSS(Extdim(X)DX (DX(F),DΩ

X)) ≥ dim(X)

and thus that dim(X) ≥ dimSS(F), meaning that F is holonomic. This finishes the proof of (i). Now(ii) is more or less a restatement of (i) combined with the fact that D2

X = id.


2.5.2. Pushforwards and pullbacks of holonomic D-modules. The second key ingredient to the proof ofTheorem 2.38 is the following special case. We will omit the other components of the proof to go intomore detail about this one.

Proposition 2.41. Let φ : X → Y be an open embedding with Y affine and X := Yf the locus ofnon-vanishing of a single equation. Then, the derived pushforward φ? preserves D(DMod(X)hol).

The proof of Proposition 2.41 will rely upon the following two key lemmas. The first of these is usedto prove the second, which is known as the b-function lemma and is of independent interest.

Lemma 2.42. Let U ⊂ X be an open set and F a DX-module. Then, if F|U is a holonomic DU -module,there exists a holonomic submodule F′ ⊂ F such that F′|U = F|U .

Proof. By Lemma 2.29, we may assume that F is coherent. Now, write G = F|U and M = H0(D(F)).Applying Theorem 2.40(i), we see that codimSS(M) ≥ dim(X), hence M is holonomic. Now, setF′ = DX(M). We claim that F′ is the desired submodule.

First, it is holonomic by Theorem 2.40. Now, there is a natural map DX(F)→ H0(DX(F)) = M, thusa natural map F′ = DX(M)→ D2

X(F) ' F. Now, because φ∆ DX ' DU φ∆ for the exact functor φ∆,we see that

F′|U ' φ∆F′ ' DU (φ∆H0(DX(F)))) ' DU (H0(φ∆DX(F))) ' DU (H0(DU (G))) ' G.

Thus, we see that applying φ∆ to the natural map F′ = DX(M) → D2X(F) ' F yields the desired

isomorphism F′|U ' F|U .

Lemma 2.43. Let X be a smooth affine variety. For f ∈ OX , let Y := Xf be the locus of non-vanishingof f with i : Y → X the inclusion. Then, for any holonomic DY -module F and any section u ∈ i+(F),there exists d(n) ∈ DX [n] and b(n) ∈ k[n] such that

d(n)(fn+1u) = b(n)fnu.

Proof. For an indeterminate λ, let Xλ := X ×kk(λ) be the extension of scalars of X to the field k(λ) and

define Yλ analogously. Then, consider the DYλ -module Fλ which is isomorphic as a OYλ -module to

fλ · k(λ)⊗kF,

with the action of DYλ given by

ξ · (fλ ·m) = fλ · λξ(f)

f·m+ fλξ(m).

It is clear that Fλ is coherent because F is; further, a good filtration FiF on F induces a good filtrationFiFλ := (FiF)λ which is evidently also good. Therefore, SS(Fλ) has the same dimension over k(λ) asSS(F) has over k, meaning that Fλ is holonomic.11

Now, let iλ : Yλ → Xλ denote the corresponding inclusion and notice that i∆λ (iλ)+(Fλ) ' Fλ isholonomic, hence by Lemma 2.42 we may find some holonomic DXλ -submodule Gλ ⊂ (iλ)+Fλ such thati∆λ (Gλ) ' Fλ. Then, we see that (iλ)+Fλ/Gλ is supported on X−Y , hence any element of (iλ)+Fλ/Gλ isannihilated by a large enough power of f . In particular, this is true for the image fλ · u of u in (iλ)+Fλ,so there exists some K such that

fλ · fKu ∈ Gλ.

Now, recall that Gλ is holonomic and hence has finite length by Corollary 2.35, meaning that the de-creasing chain of submodules

DYλ · fλ · fKu ⊃ DYλ · fλ · fK+1u ⊃ DYλ · fλ · fK+2u · · ·of Gλ must stabilize. In particular, this means that we may find some N so that fλ ·fN ∈ DYλ ·fλ ·fN+1u,

so there is some dλ ∈ DYλ such that dλ(fλ ·fN+1u) = fλ ·fNu. Then, writing dλ = p(λ)q(λ) for p(λ) ∈ DY [λ]

and q(λ) ∈ k[λ], we see that

p(λ)fλ · fN+1u = q(λ)fλ · fNu.Then, specializing to λ = n−N and taking d(n) = p(n−N) and b(n) = q(n−N), we find that

d(n)fn+1u = b(n)fnu

in i+(F), as desired.

11We note here that nowhere in our discussion of holonomic D-modules did we assume that k was algebraically closed,

so discussing the property of being holonomic over the field k(λ) is valid.


The polynomial b(n) of minimal degree for which the conclusion of Lemma 2.43 holds is called theb-function of f . We give some concrete examples.

Example 2.44. Take X = Spec(k[x]) and f = x2 + 1. Then, we may compute

1

4[(x2 + 1)∂2

x − (2n+ 1)x∂x](x2 + 1)n+1 = n(n+ 1)(x2 + 1)n,

so b(n) = n(n+ 1) is the b-function (subject to verifying that no linear b-function exists).

Example 2.45. Take X = Spec(k[x, y]) and f = x2 + y2 + 1. Then, we may compute

1

2[(x2 + 1)∂2

x + x2∂2y − 2(n+ 1)x∂x](x2 + y2 + 1)n+1 = (n+ 1)(x2 + y2 + 1)n,

so b(n) = n + 1 is the b-function. The evident asymmetry between x and y here illustrates that therecan be multiple differential operators d(n) giving a single b-function.

Let us now sketch how Lemma 2.43 implies Proposition 2.41.

Proof of Proposition 2.41. In this situation, we recall that φ? = φ+ because φ is an affine map. Wecan reduce to the case where there is a single holonomic DX -module F, and by taking an increasingholonomic filtration of F, where φ+(F) is generated by a single section u.

The first step is now to show that φ+(F) is coherent. For this, by Lemma 2.43, we see that thesequence

DY · u ⊂ DY · f−1u ⊂ DY · f−2u · · ·eventually stabilizes, hence there is some element f−Nu which generates φ+(F).

Second, to show that φ+(F) is holonomic, we note that we are in exactly the situation of the proofof Lemma 2.43. Then, take the construction of Fλ from the proof. Recall that (φλ)+Fλ was generatedby u, which by the conclusion of Lemma 2.43 lies in the DYλ -span of fλ · fKu, which is in its holonomicsubmodule Gλ. So (φλ)+Fλ is itself holonomic.

We would like to pass from this to the fact that φ+F is holonomic. For this, choose a finite numberof diλ ∈ DYλ , the vanishing locus of whose image in SymOYλ

TYλ is SS((φλ)+Fλ). Specializing to λ = n

for small enough n < 0, we obtain din ∈ DY who kill fnu and whose vanishing locus is of dimension atmost SS((φλ)+Fλ) = dim(Y ). The former holds for all n, and the latter for all but finitely many valuesof n because it holds over k(λ) and is expressed by a non-vanishing condition (and an element of k(λ)which does not vanish can only specialize to 0 for finitely many values of n).

Recall now that for small enough n < 0, fnu generates φ+F. Now, for such an n, take H to be thesubmodule generated by fnu. The din exhibit a annihilating set for grH in SymOY TY of dimensiondim(Y ), hence the singular support of H is supported on it and has dimension dim(Y ), showing that His holonomic. But we chose n so that H = φ+F, giving the desired.

2.5.3. Duality for holonomic D-modules. We now give for completeness a description of duality for thederived category of D-modules with holonomic cohomology. We omit the proof, though we will discussa particular consequence, the classification of all irreducible holonomic DX -modules on a given varietyX. As is somewhat apparent by the definition of D, this theory will be analogous to the theory ofGrothendieck duality for coherent sheaves in algebraic geometry. Let φ : X → Y be a morphism of smoothvarieties. We define functors φ! : D(DMod(X)hol) → D(DMod(Y )hol) and φ? : D(DMod(Y )hol) →D(DMod(X)hol) by

φ! = DY φ?DX and φ? = DXφ!DY .Combined with our original derived pushforward and pullback, this yields the six functors φ!, φ

!, φ?, φ?,

DX , and DY , which will be related by the following duality theorem.12

Theorem 2.46 ([Ber83, Section 3.9]). We have the following relationships between φ!, φ!, φ?, φ?, DX ,

and DY :

(i) φ! is left adjoint to φ!,(ii) φ? is left adjoint to φ?,(iii) there is a natural map φ! → φ? which is an isomorphism if φ is proper, and(iv) if φ is smooth, then φ! = φ?[2(dimY − dimX)].

12Technically speaking, we need also to show that DX , DY preserve D(DMod(X)hol). However, as we are omitting the

proof of Theorem 2.46, we do not find this necessary.


Let us see a sample application which will be quite relevant to the second half of this essay. Leti : Z → X be a locally closed affine embedding of a smooth subvariety. Then, for any OZ-coherentDZ-module F, we define its minimal extension to be

i!?F := Im(i!F → i?F)

as a DX -module. Notice that i!?F is a holonomic DX -module (and not a complex) because DX , DZ ,and i? all send holonomic D-modules to holonomic D-modules and the property of being holonomic ispreserved under taking submodules. When F is furthermore irreducible, this construction will allow us toobtain all irreducible holonomic DX -modules by the following theorem, which we again restrict ourselvesto only quoting.

Theorem 2.47 ([HTT08, Theorem 3.4.2]). Suppose that F is an irreducible OZ-coherent DZ-module.Then its minimal extension i!?F is an irreducible holonomic DX-module. Moreover, all irreducible holo-nomic DX-modules take this form.

2.6. D-modules on P1. In this subsection, we consider the toy example of D-modules on P1 using thetheory that we have developed in this section. We note in particular that P1 is the flag variety of thesemisimple algebraic group SL2, thus our discussion here will provide a link to the discussion of theBeilinson-Bernstein theorem in the second half of the essay. First, Kashiwara’s Theorem gives us thefollowing key characterization of D-modules on projective spaces.

Proposition 2.48. Let P(V ) be the projective space of a vector space V . Then, the global sectionsfunctor provides an equivalence of categories

Γ : DMod(P(V ))→ Γ(P(V ),DP(V ))−mod.

Before proving Proposition 2.48, we require some general analysis of D-modules on projective space.Define the maps j : V −0 → V and π : V −0 → P(V ), and let xi be a basis for V . The proof willrely crucially on the following lemma, which says that for a DV−0-module arising as the pullback of aDP(V )-module, the graded structure on its global sections can be recovered from the DV−0-action.

Lemma 2.49. Let F be a DP(V )-module and G := π∆F its pullback to V − 0. Then, the ith gradedcomponent of Γ(V −0,G) given by the Gm-equivariant structure is the i-eigenspace of the action of theEuler operator ξ :=

∑i xi∂i on Γ(V − 0,G).

Proof. Because G took the form π∆F, the isomorphism

φ : act∆ G ' p∆2 G

giving the Gm-equivariant structure is given as the pullback of the identity isomorphism F → F via themorphisms π act = π p2. Therefore, φ respects the DGm×(V−0)-actions on act∆ G and p∆

2 G. Recallnow that the grading on global sections of G is given as follows; an element m ∈ Γ(V −0,G) decomposesas m =

∑imi, where

φ(1⊗m) =∑i

ti ⊗mi.

Let us reinterpret this in terms of the D-module structure. Notice that DGm×(V−0) = DV−0[t, t−1, ∂t],and that because TGm is killed by p2, the action of DGm → DGm×(V−0) on p∗2F is given solely by itsleft action on OGm . We may thus compute

(6) t∂t · φ(1⊗m) =∑i

iti ⊗mi.

On the other hand, the map TGm×(V−0) → act∗ TV−0 sends t∂t to∑i xi∂i. Computing in this way,

we see that

(7) t∂t · φ(1⊗m) = φ(t∂t · (1⊗m)

)= φ

(1⊗

∑i

xi∂im).

Combining our computations (6) and (7) of the action of t∂t, we see that ξ :=∑i xi∂i acts on the ith

graded component of Γ(V − 0,G) by multiplication by i.

Proof of Proposition 2.48. We first show that Γ(P(V ),−) is exact. In view of Lemma 2.49, we may factorΓ as the composition of the functors

DMod(P(V ))π∆

→ DMod(V − 0) Γ(V−0,−)→ Γ(V − 0,OV−0)−modker(actξ)→ Γ(P(V ),DP(V ))−mod.


Here, π∆ is exact because π is flat, so it suffices to show that ker(actξ) Γ(V −0,−) is exact. For this,consider a short exact sequence

(8) 0→ F1 → F2 → F3 → 0

of DV−0-modules. Because j is an open embedding, we see that j+ coincides with the ordinarypushfoward of O-modules, hence we have that Γ(V − 0,−) ' Γ(V,−) j+. Consider now the exactsequence

0→ j+F1 → j+F2 → j+F3 → R1j+F1 → · · · .Applying the exact functor π∆, we see that the sequence π∆R1j+F1 = 0, hence R1j+F1 is supportedat 0 ∈ V . Therefore, by Kashiwara’s Theorem, we may write R1j+F1 = i+G, where i : 0 → V isthe inclusion and G is a D-module on the single point space 0. Then G is the (possibly infinite) directsum of copies k0 of the base field, and we saw in the course of the proof of Kashiwara’s Theorem thati+(k0) ' [∂1, . . . , ∂n]k ·10, where each xi annihilates the formal element 10. The action of ξ on monomials∏i ∂

αii · 10 in Γ(V, i+(k0)) is thus given by −n −

∑i αi, a strictly negative integer. In particular, this

implies that the kernel of the action of ξ on Γ(V,R1j+F1) = Γ(V, i+G) is 0. This shows that applyingthe composition ker(actξ) Γ(V,−) j+ to (8) yields an exact sequence

0→ Γ(V − 0,F1)0 → Γ(V − 0,F2)0 → Γ(V − 0,F3)0 → 0,

showing that Γ(P(V ),−) is exact.To show that Γ(P(V ),−) gives an equivalence of categories, consider the functor

Loc(M) := DP(V ) ⊗Γ(P(V ),DP(V ))

M,

where we view M and Γ(P(V ),DP(V )) as constant sheaves on P(V ). It is easy to check that Loc is leftadjoint to Γ; further, to show that they form an equivalence, it suffices by abstract nonsense to showthat that Γ(P(V ),F) = 0 implies F = 0. We will give an identical argument to this one in detail in theproof of Corollary 4.12, so we content ourself with this sketch here.

It remains now to check that Γ(P(V ),F) = 0 implies F = 0. Take an F so that Γ(P(V ),F) = 0, andconsider the graded module Γ(V − 0, π∆F) which is trivial in grade 0. An analysis similar to the onewe performed in the proof of Lemma 2.49 shows that ∂i maps Γ(V −0, π∆F)k+1 to Γ(V −0, π∆F)k

and xi maps Γ(V −0, π∆F)k to Γ(V −0, π∆F)k+1. Therefore, the action of the Euler operator ξ andthe fact that Γ(V − 0, π∆F)0 = 0 show that Γ(V − 0, π∆F)k = 0 for k > 0. Similarly, if all xi killΓ(V −0, π∆F)k for some k, then Γ(V −0, π∆F)k is supported at 0, hence 0. Applying this to thefact that Γ(V −0, π∆F)0 = 0 shows that Γ(V −0, π∆F)k = 0 for k < 0, hence Γ(V −0, π∆F) = 0.This implies that F = 0 (for instance because for any coherent submodule H ⊂ F, the twists H(n) areeventually globally generated).

Remark. Varieties X such that the global sections functor defines an equivalence of categories betweenDX -modules and Γ(X,DX)-modules are called D-affine. In Proposition 2.48, we have shown that theprojective spaces Pn are D-affine. In these terms, the following half of this essay will be devoted toshowing (via a different technique involving Lie algebra representations) that the flag variety G/B of asemisimple algebraic group is also D-affine.

Using Proposition 2.48, we may now calculate explicitly in the category of D-modules on P1. Pick twocoordinates z and w = z−1 corresponding to a cover of P1 by two copies Uz and Uw of A1. The sheaf ofdifferential operators DP1 is then defined by

Γ(Uz,DP1) = k[z, ∂z]/〈[∂z, z] = 1,

Γ(Uw,DP1) = k[w, ∂w]/〈[∂w, w] = 1〉, and

Γ(Uz ∩ Uw,DP1) = k[z, z−1, ∂z]/〈[∂z, z] = 1, [∂z, z−1] = −z−2〉,

where the restriction maps are the natural inclusion z 7→ z, ∂z 7→ ∂z on Γ(Uz,DP1) and the map

w 7→ z−1, ∂w 7→ −z2∂z

on Γ(Uw,DP1). Now, writing all monomials in k[z, z−1, ∂z] in the form zi∂jz , we find that

Γ(P1,DP1) = span(

1, zi∂jz for i ≤ j + 1, j > 0).

One may check that Γ(P1,DP1) is generated by ∂z, z∂z, and z2∂z. Suggestively, if we normalize these as

(9) e = −∂z, h = −2z∂z, f = z2∂z,


then we may check that [e, f ] = h, [h, e] = 2e, and [h, f ] = −2f . Therefore, we see that (9) definesa map of associative algebras U(sl2) → Γ(P1,DP1). It is also easy (if a bit tedious) to check thatc = 1

2h2 + h + 2fe, the Casimir element of U(sl2), is sent to zero by this map. For U(sl2), it is known

that c generates the entire center, hence we have just manually constructed a map

U(sl2)/Z(sl2) · U(sl2)→ Γ(P1,DP1).

Evidently, this map associates to each DP1-module a U(sl2)-module where Z(sl2) acts trivially. As weshall see in the sequel, this is an instance of the Beilinson-Bernstein correspondence. Now, let us identifythe U(sl2)-representations that some specific DP1 -modules correspond to.

Example 2.50. If φ : X → P1 is a closed embedding, then

φ!?(F) = Im(φ!(F)→ φ?(F)) ' φ?(F) = φ+(F)

because φ is automatically proper, hence we may apply Theorem 2.46 to see that the map φ! → φ?is an isomorphism. Therefore, Theorem 2.47 implies that φ?(F) should be an irreducible holonomicDP1-module when F is coherent and irreducible. Consider now a closed embedding φ : x → P1 of apoint x which we will assume lies in the coordinate chart Uz. Any D-module on x is simply a vectorspace, so the only interesting D-module we produce in this way is the pushfoward φ+(kx) of the trivialDx-module kx. Explicitly, it takes the form

φ+(kx) := φ.(DP1←x ⊗Ox

kx)

and thus

Γ(P1, φ+(kx)) ' Γ(x,DP1←x) ' kx ⊗OP1,x

HomOP1(ΩP1 ,DP1)x ' (DP1)x ⊗

OP1,x

kx ' [∂z]kx,

where the action of Γ(P1,DP1) is by left multiplication. Computing the actions explicitly, we see that

−∂z · ∂iz = −∂i+1z

−2z∂z · ∂iz = −2x∂i+1z + 2(i+ 1)∂iz

z2∂z · ∂iz = x2∂i+1z − 2(i+ 1)x∂iz + i(i+ 1)∂i−1

z ,

which when x = 0 corresponds to the dual Verma module M∨−2 for sl2 with lowest weight 2. When x 6= z,we observe that the resulting sl2-representation is not a lowest weight representation for the given choiceof f, h, e, which we note implicitly defined a choice of Cartan and Borel subalgebra for sl2. Indeed, if weinstead consider the mapping

e′ = −∂z, h′ = −2(z − x)∂z, f ′ = (z − w)2∂z,

then we see that

−∂z · ∂iz = −∂i+1z

−2(z − x)∂z · ∂iz = 2(i+ 1)∂iz

(z − x)2∂z · ∂iz = i(i+ 1)∂i−1z ,

meaning that the result is a lowest weight representation of sl2 for this different choice of Cartan andBorel subalgebra. We conclude this discussion by noting that the point x =∞ should correspond to themissing case of the Verma module M−2 for sl2 with highest weight −2.

3. The geometric setting

In this section, we establish some geometric preliminaries for the rest of this essay. First, we describethe flag variety of a semisimple algebraic group G and provide a construction of some equivariant vectorbundles on it. The primary sources for this are [Spr98] and [Jan03]. Second, we define the Lie algebra gand describe the induced action of g on varieties with a G-action. Finally, we discuss the Chevalley andHarish-Chandra isomorphisms for g. The sources for this portion are [Gai05], [Hum08], and [Dix77].


3.1. Preliminaries on semisimple algebraic groups. Let G be a connected, simply connected,semisimple algebraic group over an algebraically closed field k of characteristic 0. Pick a Borel sub-group B of G and a maximal torus T ⊂ B. Let U be a maximal unipotent subgroup of B, and recallthat T ' B/U .

Let Φ(G,T ) = (X∗, R,X∗, R∨) be the root datum associated to G. That is, X∗ and X∗ are abstract

abelian groups equipped with isomorphisms X∗ ' X∗(T ) and X∗ ' X∗(T ) to the lattices of charactersand co-characters of T . In particular, for λ ∈ X∗, we denote by eλ the corresponding character eλ :T → Gm. Let R and R∨ be the roots and co-roots of G, viewed as subsets of X∗ and X∗. Denote by〈−,−〉 : X∗ ⊗ X∗ → Z the perfect pairing induced from the usual pairing between X∗(T ) and X∗(T ).Our choice of B defines a choice of positive roots R+ ⊂ R. Let the corresponding sets of simple roots andco-roots be α1, . . . , αn and α∨1 , . . . , α∨n, respectively, and let ω1, . . . , ωn ⊂ X be the fundamentalweights, which lie in X∗ because G was chosen to be simply connected. Let ρ = ω1 + · · · + ωn be halfthe sum of the positive weights.

Let W = N(T )/T be the Weyl group of G, and let si be the simple reflection corresponding to αi.For w ∈W , let `(w) denote its length, which is the length of the shortest reduced decomposition for w.Denote by w0 the longest element in W .

3.2. Flag and Schubert varieties. We are now ready to introduce our basic geometric setting, theflag variety X = G/B. Let us first recall the following facts about Borel subgroups in G.

Proposition 3.1 ([Spr98, Theorem 6.2.7]). Any two Borel subgroups in G are conjugate.

Proposition 3.2 ([Spr98, Theorem 6.4.9]). For any Borel subgroup B in G, we have NG(B) = B.

For any Borel subgroup B of G, let X = G/B be the corresponding flag variety of G, interpreted asa quotient under the action of B by right translation. The following corollary allows us to describe Xwithout fixing a choice of B.

Corollary 3.3 ([DG70, XXII.5.8.3]). For each B, X = G/B is a projective variety representing thefunctor

B(S) = Borel subgroups of G×kS.13

Proof. On k-points, the correspondence is simple; it maps the point gB of X to the Borel subgroupgBg−1. That this is an isomorphism then follows from Proposition 3.2. We refer to [DG70] for the proofin the general case; however, we will only use this correspondence on k-points in this essay.

We now summarize some well-known properties of X in the following two propositions.

Proposition 3.4. The flag variety X satisfies the following properties:

(i) the projection π : G→ X gives G the structure of a principal B-bundle over X;(ii) there is an equivalence of categories QCoh(X) ' QCoh(G)B, where B-equivariance is taken with

respect to the action of B on G by right translation.

Proof. For (i), see [Spr98, Lemma 8.5.2]. For (ii), see [FGI+05, Theorem 4.46], though that result ismuch more general than is needed here.

Proposition 3.5. The fixed points of the action of g ∈ G by left translation on X are in correspondencewith Borel subgroups containing g under the identification of Corollary 3.3.

Proof. Under the identification of Corollary 3.3, notice that g ∈ G fixes g′B if and only if g normalizesg′B(g′)−1, which by Proposition 3.2, occurs if and only if g lies in g′B(g′)−1.

3.3. Equivariant vector bundles on the flag variety. We are now ready to give a fundamentalconstruction which relates the representation theory of G to the geometry of X. For any B-module V ,define the associated sheaf L(V ) on X of sections of G×B V given by

Γ(U,L(V )) = f ∈ Oπ−1(U) ⊗kV | f(g · b) = b−1 · f(g).

The following gives some basic compatibility properties of this construction.

Proposition 3.6. We have the following:

13For general S, we must take the definition of Borel subgroup given in [DG70, XXII.5.2.3]. That is, a Borel subgroup

of a group scheme G over S is a smooth subgroup B of finite type with connected fibers over S such that each geometricfiber B ×

Sks is a Borel subgroup of G ×

Sks as a linear algebraic group over ks. We include this only for completeness, as

we will only use the case where S = Spec(k).


(i) V 7→ L(V ) defines an exact functor B −mod→ QCoh(X)G;(ii) for each V , L(V ) is a vector bundle of rank dimV ;(iii) for representations V,W of B, we have L(V ⊗W ) ' L(V )⊗ L(W ).

Proof. For (i), see [Jan03, Proposition I.5.9] and the remarks in [Jan03, Section II.4.2]. For (ii) and (iii),see the remarks in [Jan03, Section II.4.1].

This construction provides a geometric way to realize induced representations from B to G. Inparticular, because the L(V ) areG-equivariant sheaves, their global sections Γ(X,L(V )) acquire a natural

G-module structure; an alternate notation for these G-modules is indGB V .When V was originally a G-module instead of only a B-module, we may characterize L(V ) more

explicitly. Indeed, consider the vector bundle V := OX ⊗k V with a G-equivariant structure inheritedfrom the action of G on both OX and V ; by this, we mean that the isomorphism

act∗OX ⊗kV ' act∗ V → p∗2V ' p∗2OX ⊗

kV

is given by

act∗OX ⊗kV → p∗2OX ⊗

kV ⊗

kOG → p∗2OX ⊗

kV,

where the first map is given by the standard isomorphism act∗OX → p∗2OX and the action map V →V ⊗

kOG and the second map is multiplication in OG. The following proposition shows that this provides

an alternate construction for L(V ).

Proposition 3.7. Let V be a G-module. There is an isomorphism of G-equivariant vector bundles

V ' L(V ).

Proof. By definition, we have that

Γ(X,L(V )) = f ∈ OG ⊗k V | f(g · b) = b−1 · f(g).

Consider the map φ[ : V → Γ(X,L(V )) defined by

φ[(v) =(g 7→ g−1 · v

),

and let φ : V → L(V ) be the corresponding map of OX -modules. It is easy to check that this map is aG-equivariant isomorphism.

We now consider an important special case. For any λ ∈ X∗, we obtain a representation k−λ of B via

B → B/U ' T e−λ→ Gm, giving rise to a family of line bundles Lλ = L(k−λ) on X. Notice that L0 = OX ,and, by Proposition 3.6(iii), Lλ ⊗ Lµ = Lλ+µ. In fact, it is known that all line bundles on X take thisform (see [FI73, Proposition 3.1]). The line bundles Lλ satisfy some well-known properties, which wesummarize below.

Proposition 3.8. We have the following:

(i) the canonical bundle ωX of X is given by L−2ρ;(ii) the G-module Γ(X,Lλ) is non-zero if and only if λ is dominant;(iii) the line bundle Lλ is globally generated if λ is dominant and ample if λ is regular.

Proof. For (i), see [Jan03, Section II.4.2]; for (ii), see [Spr98, Theorem 8.5.8]; for (iii), see [Jan03,Propositions II.4.4 and II.4.5].

Finally, we have the classical Borel-Weil theorem, which shows that we may recover the irreduciblerepresentations of G from the global sections of Lλ.14

Theorem 3.9 (Borel-Weil). For any dominant weight λ, the G-module Γ(X,Lλ)∗ is the (unique) irre-ducible representation of G with highest weight λ.

14Much more is known about the cohomologies of the Lλ. Namely, the Borel-Weil-Bott theorem shows that each Lλ

has a unique non-zero cohomology group which is isomorphic to a certain irreducible representation of G. However, this is

not strictly necessary for our main discussion, so we will not discuss it further.


3.4. Preliminaries on Lie algebras. Following [DG80], we define the Lie algebra of G as follows.Write k[ε] = k[ε]/ε2 for the ring of dual numbers and for a scheme X, write Xε = X ×

kSpec(k[ε]). For

an algebraic group G, consider the functor Lie(G) whose S points are given by

1→ Lie(G)(S)→ G(Sε)→ G(S)→ 1.

Here for a scheme S over k, by G(S) and G(Sε) we mean HomSch/k(S,G) and HomSch/k(Sε, G). Then,

we take the Lie algebra of G to be g = Lie(G)(k). Let us see what this means concretely. Noticethat G(Spec(k[ε])) = Homk(OG, k[ε]) and that G(k) = Homk(OG, k). Now, because elements of g areidentified with elements of G(Spec(k[ε])) mapping to the inclusion of the identity in G(k), we obtain anisomorphism g ' Derk(OG, k).

Suppose now that G is connected, simply-connected, and semisimple. Let h ⊂ g and b ⊂ g denote theCartan and Borel subalgebras corresponding to T and B, respectively. Let n = [b, b] so that h ' b/n.Let b− be the opposite Borel subalgebra and set n− = [b−, b−] so that g = n−⊕h⊕n. We can make theseconstructions for all points x ∈ X under the correspondence between points of X and Borel subgroups ofG; for any such x, we will denote by bx, b

−x , nx, n

−x the subalgebras corresponding to the Borel subgroup

of x in the sense of Corollary 3.3. Let U(g) denote the universal enveloping algebra of g; the precedingdirect sum decomposition yields the following well-known decomposition for U(g).

Theorem 3.10 (Poincare-Birkhoff-Witt). As a (U(n−), U(n))-bimodule, U(g) admits the decomposition

U(g) = U(n−)⊗ U(h)⊗ U(n).

Differentiating characters T → Gm and co-characters Gm → T gives rise to canonical embeddingsX∗ → h∗ and X∗ → h such that the perfect pairing 〈−,−〉 is simply the pairing between h∗ and h.Under this correspondence, note that the roots R defined earlier correspond to the weights of h on g,the positive roots R+ correspond to the weights of h on n, and the negative roots correspond to theweights of h on n−. From now on, we will identify elements of (X∗, R,X∗, R

∨) with their images underthis embedding.

We say that λ ∈ h∗ is an integral weight if it lies in X∗; in particular, because G is simply connected,we see that λ is integral if and only if 〈λ, α∨i 〉 ∈ Z for all i. Note that the integral weights λ are exactlythose which lift to characters eλ ∈ X∗(T ) of the algebraic group. We say that a weight λ ∈ h∗ is calleddominant if 〈λ, α∨i 〉 ≥ 0 for all i and regular if 〈λ, α∨i 〉 > 0 for all i. These occur if and only if λ is anon-negative (resp. positive) linear combination of fundamental weights.

3.5. Lie algebra actions on G-equivariant sheaves. Let X be a scheme equipped with a G-actionact : G ×X → X. Let us view the action map G ×X → X as a map of functors G → Aut(X), whereAut(X) is the functor

Aut(X)(R) = AutSch/k(X ×kR,X ×

kR).

This is a map of group objects, so we may consider the corresponding induced map on Lie algebras

Lie(G)→ Lie(Aut(X)).

Taking this map on k-points, we obtain a map Lie(G)(k)→ Lie(Aut(X))(k), where we have15

Lie(Aut(X))(k) = φ ∈ Aut(Xε) | φ|X = id= φ ∈ Hom(OX [ε],OX [ε]) | φ|OX = id= Derk(OX ,OX).

Thus, we have constructed a map

(10) ρ : g→ Derk(OX ,OX)

which is the result of differentiating the G-action on OX . Unraveling the construction above, we see thatρ(ξ) is given by the composition

OXact∗→ OG ⊗OX

ξ→ OX .The following proposition shows that we may perform a similar construction for any G-equivariantquasicoherent sheaf on X.

15Recall that Lie(Aut(X))(R) satisfies the exact sequence 1→ Lie(Aut(X))(R)→ Aut(X)(Rε)→ Aut(X)(R)→ 1.


Proposition 3.11. Let F be a G-equivariant quasi-coherent sheaf. Then, there is a natural map

ρF : g→ Endk(F)

such that for ξ ∈ g and f and s local sections of OX and F, respectively, we have that

(11) ρF(ξ)(f · s) = f · ρF(ξ)(s) + ρ(ξ)(f) · s,where ρ is the map of (10).

Proof. Let φ : act∗ F → p∗2F be the isomorphism giving the G-equivariant structure on F. We constructρF explicitly. For any affine open U ⊂ X and ξ ∈ g, let ρF(ξ) be given by the composition

ρF(ξ) : Γ(U,F)act∗→ Γ(G× U, act∗ F)

φ→ Γ(G× U, p∗2F) ' OG ⊗ Γ(U,F)ξ→ Γ(U,F),

where ξ acts on OG via the identification g ' Derk(OG, k) of Subsection 3.4 in the final map.Let us now check that relation (11) holds. Let e : e → G be the inclusion of the identity into G.

Tracing the image of f · s through the composition above, we see that ρF(ξ) takes it to

ρF(ξ)(f · s) = ξ(φ(act∗(f · s)))= ξ(act∗(f) · φ(act∗(s)))

= e∗(act∗(f)) · ξ(φ(act∗(s))) + ξ(act∗(f)) · e∗(φ(act∗(s)))

= f · ρF(ξ)(s) + ρ(ξ)(f) · s.

Remark. For F = OX , the map ρOX defined in Proposition 3.11 is simply the original action map ρof (10), and the condition (11) simply states that ρ(ξ) is a derivation OX → OX . Where there is noconfusion, we will abuse notation to write ρ for ρF.

Applying Proposition 3.11 for F = L, we obtain a map

ρ : g→ Endk(L)

satisfying (11). This map extends to a map U(g)→ Endk(L) with image controlled by the following.

Corollary 3.12. The action of Proposition 3.11 defines a map

φ : U(g)→ Γ(X,DX,L).

Proof. It suffices to show that the image of ρ lies in Γ(X,F1DX,L). For sections f1, f0 ∈ Γ(X,OX), wemust check that

[f1, [f0, ρ(ξ)]] = 0.

Indeed, for any local section s ∈ L, we have by repeated application of (11) that

[f1, [f0, ρ(ξ)]](s) = f1f0ρ(ξ)(s)− f1ρ(ξ)(f0s)− f0ρ(ξ)(f1s) + ρ(ξ)(f1f0s) = 0.

3.6. Chevalley and Harish-Chandra isomorphisms. In this section, we construct and character-ize two important maps associated to a semisimple Lie algebra g, the Chevalley and Harish-Chandraisomorphisms.

3.6.1. Chevalley isomorphism. Consider the space Sym(g∗)G of polynomial functions on g invariant underthe adjoint action of G on g. Recall that the Killing form (ξ, µ) 7→ Tr(adξ adµ) provides an identificationg ' g∗ which intertwines the two adjoint actions ofG on g and g∗. Consider now the induced identificationSym(g∗) ' Sym(g); because G was taken to be connected, the invariants of the G-action on Sym(g) arethe same as those of the g-action.

Given a function f ∈ Sym(g∗)G and a choice of Borel subalgebra b ⊂ g, we may ask when therestriction of f to b factors through b → b/n ' h. This restriction corresponds to taking the image off under Sym(g∗)G → Sym(b∗)H , where the result in Sym(b∗) will still be H-invariant because H actssemisimply on g∗. Let us see what this means under the identification g ' g∗. Because Tr(adξ adµ) = 0for ξ ∈ n and µ ∈ b, we see that b∗ ' g/n ' b−, hence this restriction map translates to the map

Sym(g)G → Sym(b−)H ,

given by the projection g → g/n. It is now clear that the h-invariants of Sym(b−) lie in Sym(h) andhence that the image of f in Sym(b−)H lies in Sym(h) ' Sym(h∗). Analyzing more carefully, the factthat N(H) normalizes h, we now see that f lands in the N(H)-invariants of Sym(h∗) and thus that thisconstruction defines a map

φ : Sym(g∗)G → Sym(h∗)W .

The map φ is known as the Chevalley homomorphism and by the following theorem is an isomorphism.We denote the inverse map φ : Sym(h∗)W → Sym(g∗)G.


Theorem 3.13 ([Gai05, Theorem 2.3]). The map φ is an isomorphism Sym(h∗)W ' Sym(g∗)G.

We omit the proof of Theorem 3.13 in favor of an heuristic description of the inverse map φ. For afunction f ∈ Sym(h∗)W , set g = φ(f). By the definition of the Chevalley homomorphism, for any ξ ∈ g,we may compute the value of g(ξ) by conjugating ξ into b and considering its image under the maps

b → b/n ' hf→ k. Because f was W -invariant, we see that choosing any Borel b′ containing both h

and ξ and applying the same procedure would give the same result. For regular semisimple ξ, these willcomprise all Borel subalgebras containing ξ, a fact which will come into play later.

3.6.2. Harish-Chandra isomorphism. In this subsection, we describe the Harish-Chandra isomorphism,which characterizes the possible actions of the center Z(g) of U(g). By Schur’s lemma, the action ofZ(g) on any finite-dimensional irreducible U(g)-module factors through k, meaning that Z(g) acts via amap of k-algebras χ : Z(g)→ k, which we call a central character. For any central character χ, define

U(g)χ = U(g)/U(g) · kerχ

to be the quotient of U(g) by the (two-sided) ideal generated by kerχ. Note that U(g)χ-modules aresimply U(g)-modules where Z(g) acts by χ.

We now classify the central characters χ by relating them to characters λ ∈ h∗. By Theorem 3.10,given a choice of Cartan and Borel subalgebras h ⊂ b, U(g) splits as a direct sum

(12) U(g) = U(h)⊕ (n− · U(g) + U(g) · n),

where we note that any pure tensor in U(n−) ⊗ U(h) ⊗ U(n) which does not lie in U(h) lies in at leastone of n− · U(g) or U(g) · n. Let the map

ψb : U(g)→ U(h)

to be the projection onto U(h) under this direct sum. We call the restriction of ψb to Z(g) the Harish-Chandra homomorphism relative to b.16 The following lemma justifies the name.

Lemma 3.14. The map ψb is an algebra homomorphism Z(g)→ U(h).

Proof. This results from a careful analysis of Z(g) using Theorem 3.10. We claim that Z(g)∩ (n−U(g) +U(g)n) ⊂ U(g)n ∩ n−U(g). This would imply that ψ is a map of algebras, as for i = 1, 2, if we hadzi = hi + gi with hi ∈ U(h) and gi ∈ U(g)n ∩ n−U(g), then the decomposition

z1z2 = h1h2 + (h1g2 + g1h2 + g1g2)

would satisfy h1h2 ∈ U(h) and h1g2 + g1h2 + g1g2 ∈ U(g)n ∩ n−U(g).It remains to prove the claim. It is evidently symmetric in n and n−, so it suffices to show that

Z(g) ∩ (n−U(g) + U(g)n) ⊂ U(g)n. For this, write any element z ∈ Z(g) in the PBW ordering andconsider the adjoint action of h on z. Then, take any monomial

z′ = e−αi1 · · · e−αik gfαj1 · · · fαjk′of z with g ∈ U(h) and the αi simple roots. Then, we see that the adjoint action of h on z′ is viaαj1 + · · ·+ αjk′ − αi1 − · · · − αik , which must vanish, meaning that either k = k′ = 0 or k, k′ > 0, givingthe claim.

Viewing Sym(h) as the space of polynomial functions on h∗, we may interpret ψb as a map

h∗ → MaxSpec(Z(g)),

where the central characters χ are in bijection with MaxSpec(Z(g)). For λ ∈ h∗, denote by χbλ the central

character corresponding to λ relative to b. We may immediately understand the following importantconcrete example of the action of Z(g) through a central character.

Corollary 3.15. For any weight λ ∈ h∗, Z(g) acts via χbλ on the U(g)-module

Mbλ := U(g) ⊗

U(b)kλ,

which is known as the Verma module of highest weight λ.

Proof. This follows immediately from the fact that Z(g)∩ (n−U(g) +U(g)n) ⊂ U(g)n, as n acts triviallyin kλ, meaning that the action of Z(g) can be through only the part preserved by ψb.

16While the definition of ψb depends on the choice of both h and b, we write ψb instead of ψh,b because the latter is

somewhat cumbersome.


Thus far, we have considered what we call the Harish-Chandra homomorphism relative to b, a mapψb : Z(g)→ Sym(h) which may (and does) depend on the embedding h ⊂ b ⊂ g we chose when applyingthe PBW decomposition. This dependency is not strictly necessary, as both Z(g) and h can be definedabstractly without reference to a specific choice of Cartan or Borel subalgebra. More concretely, whatwe have done so far does not allow us to compute the action of Z(g) on Verma modules

U(g) ⊗U(bx)

kλ

relative to arbitrary Borel subalgebras. As we have hinted in our nomenclature thus far, it is possibleto modify the construction of ψb to obtain a map ψ : Z(g)→ U(h) that is independent of the choice ofCartan subalgebra. Indeed, define the map ψ, which we will call the Harish-Chandra homomorphism, tobe the composition

Z(g)ψb→ U(h)

ξ 7→ξ−ρb(ξ)→ U(h),

where the map U(h) → U(h) is the one induced by mapping h → U(h) via ξ 7→ ξ − ρx(ξ). Here, wewrite ρb for the longest positive root relative to b to emphasize that the choice of ρb ∈ g depends on thechoice of h ⊂ b ⊂ g. We see that ψ then admits the following nice characterization, which is known asthe Harish-Chandra isomorphism.

Theorem 3.16 ([Dix77, Theorem 7.4.5]). The map ψ is an isomorphism Z(g) → Sym(h)W which isindependent of the choice of h ⊂ b ⊂ g.

We omit the proof of Theorem 3.16 and instead discuss some consequences. Notice that ψ gives adifferent map

(13) h∗ → MaxSpec(Z(g)).

For λ ∈ h∗, denote now by χλ the central character corresponding to λ. We then have the followingcharacterization of the space of central characters.

Corollary 3.17. We have the following:

(i) every central character χ lies in the image of the map (13), and(ii) χλ = χµ if and only if there is some w ∈W such that λ = w(µ).

Proof. This follows formally from the Harish-Chandra isomorphism and the following standard fact aboutquotients of affine schemes by actions of finite groups (see [Har92, Lecture 10]). Let Spec(A) be an affinevariety over k equipped with the action of a finite group G. Then, the map MaxSpec(A)→ MaxSpec(AG)is the quotient map for the G-action on MaxSpec(A).

Applying this for A = Sym(h) and G = W acting on Sym(h), (i) follows because the quotient map issurjective, and (ii) follows because the fibers of the quotient map are exactly W -orbits.

This provides a significantly more flexible perspective from which to compute the action of Z(g) onVerma modules corresponding to arbitrary Borel subalgebras. In particular, we have the following twocomputations which will be crucial later on.

Corollary 3.18. For any weight λ ∈ h∗ and any Borel subalgebra bx with corresponding longest positiveroot ρx := ρbx , Z(g) acts on the Verma module

Mbxλ := U(g) ⊗

U(bx)kλ

of highest weight λ relative to bx via χλ+ρx .

Proof. This is a direct translation of Corollary 3.15 into the language of Theorem 3.16.

Remark. By Corollary 3.18, we see that the action of Z(g) on the Verma module Mbxλ−ρx is given by

χλ, which is independent of the choice of bx.

Corollary 3.19. For any weight λ ∈ b∗ and any Borel subalgebra bx with corresponding longest positiveroot ρx, Z(g) acts on the right U(g)-module

λMbx := kλ ⊗

U(bx)U(g)

via the central character χλ−ρx .


Proof. Recall that there is a standard anti-involution τ : U(g) → U(g) induced by the map ξ 7→ −ξ ong. This map gives rise to an isomorphism U(g) ' U(g)op that fixes Z(g) pointwise and interchanges bxand b−x for each x (see [Dix77, Proposition 2.2.17]). Viewing λM

bx as a left U(g)-module under thiscorrespondence, we see that it identifies with

Mb−xλ = U(g) ⊗

U(b−x )

kλ,

where b−x is the Borel subalgebra opposite to bx. By Corollary 3.18, Z(g) acts via χλ+ρ−x= χλ−ρx on

Mbxλ , giving the desired conclusion.

For context, we translate the content of Corollary 3.17 into our original situation of a fixed chosenBorel subalgebra b. Define the dotted action of W on h∗ by

w · λ = w(λ+ ρb)− ρb

for any λ ∈ h∗. Then, we see that χbλ = χb

µ if and only if λ and µ are in the same W -orbit under thedotted action.

4. Beilinson-Bernstein localization

In this section, we may now state and prove the main result of this essay, the Beilinson-Bernsteinlocalization theorem. This result relates the category of modules over a sheaf DλX of twisted differentialoperators on the flag variety X of a semisimple algebraic group G to the category of representations ofits Lie algebra g acting with a given central character. The proof divides into two main steps. First, weidentify the global sections of DλX with a quotient of the universal enveloping algebra of g. The proofpasses to the associated graded rings and crucially involves Kostant’s theorem on the ring of regularfunctions of the cone of nilpotent elements in g. We give a proof of this using the Springer resolution ofthe nilpotent cone.

Second, we show that the category of DλX -modules is equivalent to the category of modules over theglobal sections Γ(DλX). The main idea of the proof is to tensor a DλX -module with known vector bundlesassociated to G-representations to create a result on which the global sections functor has nice properties.A splitting trick of [BB81] then allows us to pass back to the original module.

We must discuss a fine point of notational convention before we proceed. Throughout this section, wewill assume that we have a fixed choice of a distinguished Borel subgroup B ⊂ G and the correspondingBorel subalgebra b ⊂ g, which will correspond to the set of positive roots R+. The positivity, dominance,and regularity of weights will be taken with respect to this choice.

We draw from a number of sources for this section. For preliminary constructions involving twisteddifferential operators and Lie algebroids, we refer to the original papers [BB81], [BB93], and [Kas89].Our treatment of Kostant’s theorem and the geometry of the nilpotent cone is based primarily on that of[Gai05], though we appeal to [BL96], [Kos59], and [HTT08] to fill in some details of the proofs. Finally,for the proof that the global sections functor gives an equivalence of categories, we follow the originalproof of [BB81], though we consulted [HTT08] and [Kas89] to understand some of the more subtle points.

4.1. Twisted differential operators. Before we state the Beilinson-Bernstein correspondence, we mustfirst introduce a generalization of the sheaf of differential operators.

Definition 4.1. For a smooth algebraic variety X, a sheaf of twisted differential operators on X is asheaf D of OX -algebras on X equipped with a filtration FiDi≥0 such that

(i) the inclusion OX → D gives an isomorphism OX ' F0D,(ii) the natural map SymOX F1D/F0D → grD is an isomorphism of OX -algebras, and(iii) the map F1D/F0D → TX defined by

ξ 7→(f 7→ ξf − fξ

)is an isomorphism.

Remark. We interpret condition (iii) in Definition 4.1 as follows. For sections ξ ∈ F1D and f ∈ OX 'F0D, the commutativity of grD ensures that f 7→ ξf − fξ ∈ F0D ' OX .

Definition 4.2. For a sheaf of twisted differential operators D, a D-module on X is a quasicoherentsheaf equipped with a D-action compatible with the OX -action via the inclusion OX → D.


By Propositions 2.4 and 2.5, we see that DX is a sheaf of twisted differential operators on X. Thisis an instance of the following more general construction. For L a line bundle on X, define a L-twisteddifferential operator on X of order at most n to be a k-linear map d : L → L such that for sectionsf0, f1, . . . , fn ∈ OX , we have

[fn, [fn−1, [· · · , [f0, d]]]] = 0

as a k-linear map L→ L, where a section f ∈ OX gives rise to a k-linear map L→ L via the OX -actionon L. These operators form a sheaf of rings DX,L on X, which we call the sheaf of L-twisted differentialoperators on X.17 Notice that a OX -twisted differential operator on X is simply what we previouslytermed a differential operator on X, and hence that DX,OX = DX . In analogy with DX , we endow DX,Lwith the order filtration FnDX,L. By the following proposition, this construction results in a sheaf oftwisted differential operators.

Proposition 4.3. For any line bundle L on X, DX,L is a sheaf of twisted differential operators on X.

Proof. We check the conditions in Definition 4.1, though in a different order. For (i), an element ofF0DX,L is simply an OX -linear map L → L, so F0DX,L ' OX because L was a line bundle. For (ii)and (iii), it is enough to check this locally, in which case L|U ' OU and the statements follow from thecorresponding ones for DU (Propositions 2.4 and 2.5).

By construction, L is a DX,L-module, and for L = OX , the DX,L-module structure induced on OXis simply the natural DX -module structure of Example 2.11.

4.2. Lie algebroids and enveloping algebras on the flag variety. Let us now restrict to the casewhere X = G/B is the flag variety of a connected, simply connected, semisimple algebraic group G. Wenow describe a way to understand the notion of a sheaf of g-modules on X. The key advantage of thisnotion over simply considering the g-module alone will be the ability to consider all Borel subalgebrasand subgroups of g and G at once.

Define the Lie algebroid g of the Lie algebra g to be a sheaf of Lie algebras isomorphic to OX ⊗k g asa OX -module and with Lie bracket [−,−] : g⊗k g→ g extending the bracket on g such that for f ∈ OXand ξ1, ξ2 ∈ g, we have

[ξ1, f · ξ2] = f [ξ1, ξ2] + ρ(ξ1)(f)ξ2.

Then, define the universal enveloping algebra U(g) to be the enveloping algebra of g. That is, it is a sheafof OX -algebras isomorphic to OX ⊗k U(g) as a quasi-coherent OX -module with multiplicative structuregiven by extending the structure on U(g) subject to the twisting relation

[ξ, f ] = ρ(ξ)(f)

for ξ ∈ g and f ∈ OX . Recall here that ρ : g → TX is the map (10) induced by the G-action on X; itnaturally extends to a map of OX -algebras

(14) ρ : U(g)→ DX .For any G-module V , let V := OX ⊗k V be the G-equivariant sheaf associated to V as a B-module.

Then, V acquires the structure of a U(g)-module by differentiating the action of G; explicitly, ξ ∈ g,viewed as a local section of U(g), acts by

ξ · (f ⊗ v) = ρ(ξ)(f)⊗ v + f ⊗ ξ · von a local section f ⊗ v of V. When V is only a U(g)-module, this action still exhibits V as a U(g)-module, though it is no longer a G-equivariant sheaf. Conversely, for any U(g)-module M, its localsections Γ(U,M) acquire the structure of U(g)-modules via the inclusion U(g) → Γ(U,OX)⊗k U(g). Fortwo U(g)-modules M1 and M2, we may equip M1 ⊗M2 with the structure of a U(g)-module by lettinga local section ξ ∈ U(g) act via

ξ · (m1 ⊗m2) = ξ ·m1 ⊗m2 +m1 ⊗ ξ ·m2

on a local section m1 ⊗m2 ∈ M1 ⊗M2. It is easy to check that this defines a valid action of U(g) andthat the operation M1 ⊗− is functorial.

Define the center Z(g) of U(g) by applying the corresponding construction for Z(g) ⊂ U(g) and viewZ(g) as a subsheaf of U(g). Further, restricting the map (14) to g ⊂ U(g), we define

b := ker(ρ : g→ TX)

17If the line bundle L is replaced by a vector bundle M, we may still define a sheaf of M-twisted differential operators.

However, it will not be a sheaf of twisted differential operators in the sense of Definition 4.1.


and n := [b, b]. Observe that b and n are subsheaves of g. We may describe b explicitly as

b = f ∈ g | f(x) ∈ bx for all x ∈ X,where here bx ⊂ g is the Borel subalgebra corresponding to x ∈ X. As a result, we see that

n = f ∈ g | f(x) ∈ nx for all x ∈ X,

and that b/n ' h := OX ⊗kh. For a map λ : b → OX and a U(g)-module M, we say that U(b) ⊂ U(g)

acts via λ on M if the action of U(b) factors through λ. If λ ∈ h∗ is a weight, we denote the map

b→ b/n ' hλ→ OX it induces also by λ.18

4.3. A family of twisted differential operators. Recall from Subsection 3.3 that for each charactere−λ ∈ X∗(T ) we have a G-equivariant line bundle Lλ given by the corresponding one-dimensionalrepresentation of B. Specializing the previous construction, we note that DX,Lλ is a sheaf of twisteddifferential operators on X. This construction for DX,Lλ requires λ to be a integral weight (so that

it could lift to a character of T ). We now extend it to a family DλX of sheaves of twisted differentialoperators parametrized by any weight λ ∈ h∗.19

Let λ ∈ h∗ be an arbitrary weight. We construct a sheaf DλX as follows. Consider the map ρ : b→ OXgiven on each fiber by taking the action of ρx on bx. Define the right ideal Iλ(g) ⊂ U(g) generated bythe local sections

ξ − (ρ− λ)(ξ) ∈ U(g)

for ξ ∈ b. The following lemma shows that it is possible to quotient by Iλ(g).

Lemma 4.4. For λ ∈ h∗ an arbitrary weight, Iλ(g) is a two-sided ideal.

Proof. It suffices for us to show that [b, g] ⊂ b. But this follows because the map ρ of (14) was a map of

OX -algebras, hence for any ξ ∈ b and µ ∈ g, we have

ρ([ξ, µ]) = ρ(ξ)ρ(µ)− ρ(µ)ρ(ξ) = 0,

hence [ξ, µ] ∈ b.

Remark. Lemma 4.4 illustrates the essential role that sheaves play in the theory. In particular, a naivedefinition of an analogous left ideal Iλ(g) ⊂ U(g) generated by ξ − (ρ − λ)(ξ) for ξ ∈ b does not resultin a two-sided ideal. The problem is that b ⊂ g is not globally the kernel of the map g→ Γ(X,TX). By

considering instead the sheaf b, we are able to detect more local behavior.

Now, define the sheaf of OX -algebras DλX to be the quotient

DλX := U(g)/Iλ(g),

and define the maps

(15) Φλ : U(g)→ DλXand

(16) φλ : U(g)→ Γ(X,DλX)

to be the projection map and the map given by composing the action of Φλ on global sections with thenatural inclusion U(g) → Γ(X,OX)⊗k U(g), respectively.

Proposition 4.5. For any weight λ ∈ h∗, DλX is a sheaf of twisted differential operators.

Proof. We check each property in turn. Condition (i) is obvious. For condition (iii), note that when

restricted to F1U(g)/F0U(g) ' g, the image of the inclusion Iλ(b) → U(g) is simply b, hence the

isomorphism follows from the exact sequence 0 → b → g → TX → 0. Finally, for condition (ii), note

that the image of Iλ(g) in FiU(g)/Fi−1U(g) ' SymiOX g is given by b · Symi

OX g, hence we see that

gri U(g)/Iλ(g) ' SymiOX TX ,

as needed, where we are using the fact that g/b ' TX .

Corollary 4.6. For any weight λ ∈ h∗, U(b) acts via ρ− λ on DλX .

18We refer the reader to [BB93] for more information about constructions of this sort and their generalizations.19There is a more general theory of twisted differential operators, for which we refer the reader to [BB93]. However, we

will in this essay only be concerned with sheaves of twisted differential operators of the form DλX .


Proof. This follows because the action of U(b) is by applying Φλ and the multiplying on the left.

Now, let λ ∈ h∗ be an integral weight. Because Lλ is G-equivariant, by Corollary 3.12 we have amap φλ : U(g) → Γ(X,DX,Lλ−ρ). Together with the inclusion OX → DX,Lλ−ρ , this defines a map ofOX -algebras Ψλ : U(g) → DX,Lλ−ρ . We would like to show that Ψλ realizes DX,Lλ−ρ as exactly the

quotient U(g)/Iλ(g) of U(g), which would mean that our new family DλX extends the family DX,Lλ−ρ .

Lemma 4.7. Let λ ∈ h∗ be an integral weight. Then, the map Ψλ : U(g) → DX,Lλ−ρ defines anisomorphism

DλX → DX,Lλ−ρ ,

where here we take ρ corresponding to the choice of B used to construct Lλ−ρ.

Proof. First, we claim that the map U(g) → DX,Lλ−ρ kills Iλ(g). For this, it suffices to note that for

each x ∈ X, the corresponding Borel subalgebra bx acts on the fiber Lλx by eρx−λ (this is ρx instead of ρbecause we are on bx instead of b). The resulting map is an isomorphism by the following commutativediagram on the level of the associated graded

SymOX TX grDX,Lλ

grDλX ,

6 -

where the top and left maps are isomorphisms by Propositions 4.3 and 4.5.

4.4. The localization functor and the localization theorem. Given a DλX -module M on X, themap (16) endows its global sections Γ(X,M) with the structure of a U(g)-module. Conversely, given aU(g)-module M , the map φλ of (16) allows us to construct the DλX -module

Locλ(M) := DλX ⊗U(g)

M,

where U(g) is viewed as a (locally) constant sheaf of algebras on X, M is viewed as the corresponding(locally) constant sheaf of modules over U(g), and the map U(g) → DλX is induced by φλ.20 We callLocλ(M) the localization of M and Locλ a localization functor. The following shows that it is left adjointto the global sections functor Γ := Γ(X,−).

Proposition 4.8. The functors Locλ : U(g)−mod DλX −mod : Γ are adjoint.

Proof. This follows from the following chain of natural isomorphisms of bifunctors

HomDλX (Locλ(−),−) = HomDλX (DλX ⊗U(g)−,−) ' HomU(g)(−,HomDλX (DλX ,−)) ' HomU(g)(−,Γ(X,−)).

To further characterize the relationship between Locλ and Γ, we must analyze φλ more carefully. Bythe following theorem, whose proof we defer to Subsection 4.5, φλ allows us to identify Γ(X,DλX) withU(g)χ for χ = χ−λ.

Theorem 4.9. The map φλ : U(g)→ Γ(X,DλX) defines an isomorphism

U(g)χ−λ ' Γ(X,DλX).

An immediate consequence of Theorem 4.9 is the following.

Corollary 4.10. If M is a U(g)χ-module for some χ 6= χ−λ, then Locλ(M) = 0.

Proof. Take some z ∈ Z(g) with χλ(z) 6= χ(z). For any local section ξ ⊗m of Locλ(M), we have

ξ ⊗m =1

χ−λ(z)− χ(z)· ξ · (z − χ(z))⊗m =

1

χ−λ(z)− χ(z)ξ ⊗ (z − χ(z)) ·m = 0,

where the first equality follows from Theorem 4.9. We conclude that Locλ(M) = 0.

20Equivalently, the map factors as U(g)→ U(g)→ DλX .


Thus, we see that Locλ kills the subcategories U(g)χ −mod of U(g)−mod unless χ = χ−λ. On theother hand, by Theorem 4.9, Γ is a functor DλX −mod→ U(g)χ−λ −mod.21 The following theorem andits corollary, which is the main result of this essay, show that when λ is regular, these functors give anequivalence of categories. We defer the proof of Theorem 4.11 to Subsection 4.7.

Theorem 4.11. Let λ ∈ h∗ be an arbitrary weight. Then, we have the following:

(i) if λ is dominant, then Γ : DλX −mod→ U(g)χ−λ −mod is exact, and

(ii) if λ is regular, for Γ : DλX −mod→ U(g)χ−λ −mod, if Γ(X,F) = 0, then F = 0.22

Corollary 4.12 (Beilinson-Bernstein localization). If λ is regular, then

Locλ : U(g)χ−λ −mod DλX −mod : Γ

is an equivalence of categories.

Proof. This follows formally from Theorem 4.11. Indeed, by Proposition 4.8, Locλ and Γ are adjoint,hence it suffices to check that the unit and counit maps are isomorphisms.

First, consider the map Locλ Γ → id. It becomes an isomorphism after application of Γ by theadjunction. However, Γ is exact and conservative by Theorem 4.11, which implies that the original mapis an isomorphism. Indeed, for any M ∈ U(g)χλ −mod, taking the exact sequence

0→ K → Locλ(Γ(X,M))→M→ C → 0

gives an exact sequence

0→ Γ(X,K)→ Γ(X,Locλ(Γ(X,M)))→ Γ(X,M)→ Γ(X,C)→ 0

with Γ(X,Locλ(Γ(X,M))) ' Γ(X,M), which shows that Γ(X,K) = Γ(X,C) = 0, hence C = K = 0.Next, consider the map id→ Γ Locλ. For any M ∈ U(g)χ−λ −mod, take a free resolution

U(g)Iχ−λ → U(g)Jχ−λ →M → 0

for M for some index sets I and J . Now, by Theorem 4.11 Γ Locλ is right exact as the composition ofan exact functor and a left adjoint, hence we obtain the commutative diagram of right exact sequences

U(g)Iχ−λ- U(g)Jχ−λ

- M - 0

U(g)Iχ−λ

∼

?- U(g)Jχ−λ

∼

?- Γ(X,Locλ(M))

?- 0

where the first two columns are isomorphisms. By the five lemma, M → Γ(X,Locλ(M)) is an isomor-phism, completing the proof.

At first glance, Corollary 4.12 applies only to weights λ lying in the principal Weyl chamber µ ∈h∗ | 〈µ, α∨i 〉 > 0. However, because W acts simply transitively on the Weyl chambers, for any weightλ which avoids the root hyperplanes λ ∈ h∗ | 〈µ, α∨i 〉 = 0, we may find a unique w ∈ W such thatµ = w(λ) is regular. Therefore, Corollaries 3.17 and 4.12 give equivalences of categories

U(g)χ−λ −mod = U(g)χ−µ −mod ' DµX −mod.

Remark. We summarize here the relevant notational conventions underlying the statement of Corollary4.12. Recall that we chose the positive roots R+ to correspond to the Borel subgroup B, and the G-equivariant line bundle Lλ to correspond to the character e−λ rather than eλ (we make this choice sothat dominant and regular λ correspond to globally generated and ample Lλ in Proposition 3.8). Inparticular, the b-action on local sections of Lλ is via the weight −λ rather than λ. We emphasize,however, that under our definition of

DλX = U(g)/〈ξ − (ρ− λ)(ξ)〉U(g),

21This fact places Locλ in analogy with the following situation for schemes which explains why it is known as the

localization functor. For X a scheme, there is a left adjoint LocX : Γ(X,OX)−mod→ QCoh(X) to the functor of globalsections given by LocX(M) := OX ⊗

Γ(X,OX )M . For quasicoherent sheaves, LocX is an equivalence if and only if X is affine;

however, for D-modules, the key consequence of the Beilinson-Bernstein correspondence is that the localization functor is

an equivalence more frequently.22Recall that a functor F is called conservative if, for F (f) an isomorphism, f is an isomorphism. When F is an exact

functor between abelian categories, the condition of (ii) implies that F is conservative.


the sheaf DλX corresponds to DX,Lλ−ρ . In particular, we have that DρX ' DX . As we shall see in theproof of Proposition 4.13, this shift is necessary to ensure that Z(g) acts by the same central characteron all fibers of DλX . Finally, we note the action of Z(g) on the global sections of DλX -modules is via χ−λ(rather than χλ).

4.5. The map U(g) → Γ(X,DλX). In this subsection, we prove Theorem 4.9 subject to an algebraicstatement related to the geometry of the cone of nilpotent elements in g, which we defer to the nextsubsection. We begin by checking that the statement of the theorem makes sense.

Proposition 4.13. The map φλ : U(g)→ Γ(X,DλX) factors through U(g)χ−λ .

Proof. We recall that the map φλ was obtained by considering

U(g) → Γ(X,U(g))→ Γ(X,DλX).

Letting J−λ(g) ⊂ U(g) be the ideal generated by z − χ−λ(z) for z ∈ Z(g), it suffices for us to show thatthe composition

J−λ(g) → U(g)→ DλXis zero. In fact, it suffices to check this on fibers. For each x ∈ X, by right exactness of kx ⊗−, we seethat

kx ⊗OXDλX = kx ⊗

OXDλX ' U(g)/(kx ⊗

OXIλ(g)),

where we observe that kx ⊗OXIλ(g) is the right ideal of U(g) generated by ξ − (ρx − λ)(ξ) for ξ ∈ bx.23

In other words, we have shown that

kx ⊗OXDλX ' kρx−λ ⊗

U(bx)U(g).

By Corollary 3.19, the action of Z(g) on this right U(g)-representation is by χρx−λ−ρx = χ−λ, henceJ−λ(g)x dies on each fiber, as needed.

Remark. The proof of Proposition 4.13 allows us to identify

Γ(X, kx ⊗OXDλX) ' kρx−λ ⊗

U(bx)U(g)

as a right Verma module which under the isomorphism between U(g) and U(g)op will transform to adual Verma module.

By Proposition 4.13, φλ induces a map U(g)χ−λ → Γ(X,DλX), which we will denote by ψλ. We wouldlike to show that ψλ is an isomorphism. For this, we will require the following algebraic consequence,Proposition 4.14, of a theorem of Kostant, Theorem 4.19. Kostant’s theorem will require some additionalgeometric background to state, so for now we describe only Proposition 4.14 and how it may be used toprove Theorem 4.9. In the next subsection, we will state and prove Kostant’s theorem and then use itto derive Proposition 4.14.

Recall now that the G-action on X gives rise to a map ρ : g→ Γ(X,TX) of (10) given by differentiatingthe action. This map induces a map Sym(g)→ Γ(X,SymOX (TX)), which we may characterize as follows.

Proposition 4.14. Let Sym(g)G+ denote the elements of positive grade in Sym(g)G. The map

Sym(g)/Sym(g) · Sym(g)G+ → Γ(X,SymOX TX)

is an isomorphism.

Modulo the following lemma, we are now ready to prove Theorem 4.9.

Lemma 4.15. The inclusion Z(g)→ U(g) gives rise to an isomorphism

grZ(g) ' Sym(g)G.

Proof. By definition, we have for each i the short exact sequence

0→ Fi−1U(g)→ FiU(g)→ Symi(g)→ 0

induced by the order filtration on U(g). Viewing this as a sequence of g-representations, it splits bycomplete reducibility. We may therefore apply the functor of G-invariants (which coincides with thefunctor of g-invariants) to obtain an exact sequence

0→ Fi−1Z(g)→ FiZ(g)→ Symi(g)G → 0

which induces the desired isomorphism grZ(g)→ Sym(g)G.

23We emphasize here that Iλ(g)x is not a left ideal of U(g).


Proof of Theorem 4.9. Equip U(g)χ−λ with a filtration inherited from the filtration of U(g) by order. Bydefinition φλ respects the filtration, so it suffices for us to show that the induced map

grφλ : grU(g)χ−λ → gr Γ(X,DλX)

is an isomorphism. Our approach for this will be to construct a chain of maps

Sym(g)/Sym(g) · Sym(g)G+ grU(g)χ−λ → gr Γ(X,DλX) → Γ(X, grDλX) ' Γ(X,SymOX TX),

with the first and last map surjective and injective, respectively, whose composition is the map ofProposition 4.14. Because it is an isomorphism, each of the maps will be an isomorphism, giving theclaim.

We now construct the two desired maps. For the first map, we have a natural surjective map Sym(g)→grU(g)χ−λ ; it suffices to check that it kills Sym(g)G+. But recall from Lemma 4.15 that grZ(g) 'Sym(g)G, hence for any element f ∈ Sym(g)G+, we see that the image of f in Sym(g) lies in (grZ(g))+ ⊂grU(g). But χ−λ sends Z(g)→ k ' F0U(g), hence this image dies in grU(g)χ−λ . For the second map,we have for all i a short exact sequence of sheaves

0→ Fi−1DλX → FiDλX → griDλX → 0,

hence applying the left-exact functor Γ(X,−) yields an injection

gri Γ(X,DλX) = FiΓ(X,DλX)/Fi−1Γ(X,DλX) → Γ(X, griDλX),

where we are using the fact that FiΓ(X,DλX) = Γ(X,FiDλX). It is clear that the resulting mapgr Γ(X,DλX) → Γ(X, grDλX) respects the ring structure, hence this gives the desired map.

4.6. The geometry of the nilpotent cone and Kostant’s Theorem. It now remains for us to proveProposition 4.14. We will reinterpret the statement in terms of the geometry of the nilpotent cone N ofnilpotent elements in g. Therefore, let us begin by considering the geometric situation more carefully.We note that N is in fact a variety, as the condition that adx is a nilpotent linear map g → g is a

polynomial one. Now, recall that in Subsection 4.2, we defined the vector bundles g, b, and n on Xwhose fibers over x ∈ X were g, bx, and nx, respectively. Let h//W := Spec(Sym(h∗)W ) be the quotientof h by the W -action, and consider the diagram

(17)

SymX b - h

g?

- h//W?

where the maps SymX b → g and SymX b → h are given by the inclusion b → g and the projection

b/n → h and the map g → h//W is given by the Chevalley isomorphism. We now give some initialcharacterizations of (17).

Lemma 4.16. The diagram (17) commutes, the variety h//W is smooth, and the maps h→ h//W andg→ h//W are flat.

Proof. For commutativity, we check that the corresponding maps of structure sheaves agree. We wouldlike to show that

SymOX b∗ Sym(h∗)

Sym(g∗)

6

Sym(h∗)W

6

commutes, for which it suffices to note that for f ∈ Sym(h∗)W , the image of f tracing up and to the left

is a function on SymX b which factors through the map

SymX b→ SymX hf→ A1.

On the other hand, the image of f tracing left and up is a function on SymX b which assigns to a pair(bx, ξ) with ξ ∈ bx the value of the corresponding function φ(f) at ξ, where φ is the map of the Chevalley

isomorphism. But by Theorem 3.13, such a map factors through the projection bx → bx/nx → hf→ A1,

hence the two resulting maps on SymX b agree, as needed.


The fact that h//W is smooth follows from the fact that it is affine as the quotient of a finite reflectiongroup acting on a vector space (see [Hum90, Theorem 3.5]). Now, h → h//W is a map with fibers ofdimension 0 with h Cohen-Macaulay and h//W regular, hence it is flat.

Finally, to show that the map g → h//W induced by the Chevalley isomorphism is flat, we mustshow that Sym(g∗) is flat over Sym(h∗)W ' Sym(g∗)G. We give a sketch here of the very elegantargument presented in [BL96, Theorem 1.2]. Fix a choice of h ⊂ b ⊂ g, which gives rise to a canonicaldecomposition g∗ ' h∗ ⊕ (g/h)∗. Place an alternate filtration F ′i Sym(g∗) on Sym(g∗) so that

F ′i Sym(g∗) = Sym((g/h)∗) · Fi Sym(g∗),

where Fi is the standard filtration. We then check that under this filtration gr Sym(g∗) ' Sym(h∗) ⊗Sym((g/h))∗, where the grading on the second term is on Sym(h∗) only. Let us now consider gr Sym(g∗)as a gr Sym(g∗)G-algebra with this modified filtration. Now, the image of gr Sym(g∗)G in gr Sym(g∗) isthe same as its image after restriction to Sym(h∗). But by the Chevalley isomorphism, this is simplygr Sym(h∗)W . We see therefore that the map gr Sym(g∗)G → gr Sym(g∗) factors through

gr Sym(g∗)G ' Sym(h∗)W → Sym(h∗) → Sym(h∗)⊗ Sym((g/h))∗ ' gr Sym(g∗),

where we showed that Sym(h∗) was flat over Sym(h∗)W . The result then follows from the following twofacts. First, a module over a graded algebra is flat if and only if it is free ([Eis95, Exercise 6.11]), whichimplies that Sym(h∗) is free over Sym(h∗)W , hence gr Sym(g∗) is free over gr Sym(g∗)G. Second, if grMis a free grA-module, then M is a free A-module ([BL96, Lemma 4.2]); this follows from noting that anisomorphism (grA)n → grM comes from a map An → M with a special filtration on An. This impliesthat Sym(g∗) is free over Sym(g∗)G, as needed.

Now, we say that ξ ∈ g is regular if its centralizer zg(ξ) = µ ∈ g | adξ(µ) = 0 has minimal dimension.The locus of regular elements is determined by the non-vanishing of a determinantal ideal, hence formsa Zariski-dense open set in g, which we denote by greg. Recall that this minimal dimension is called therank of g and for semisimple Lie algebras the rank is given by dim h (see for instance [Ser01, CorollaryIII.5.2]). Importantly, the following proposition shows that the Chevalley map φ is smooth on greg.

Proposition 4.17. The restriction of the Chevalley map φ to greg is smooth.

Proof. We refer the reader to [HTT08, Theorem 10.3.7] for the proof, which proceeds by considering thedifferential form det(Ω) on g defined by Ωa(x, y) = (a, [x, y]), where (−,−) is the Killing form. This formturns out to be non-vanishing exactly on greg, and its Hodge star may be computed to be the Jacobianof the Chevalley map, giving the result.24

This allows us to give a characterization of (17). Let us write g′ := g ×h//W

h.

Lemma 4.18. Let (SymX b)reg be the preimage of greg in SymX b. Then, the following diagram isCartesian.

(SymX b)reg - h

greg?

- h//W?

Proof. Denote by g′reg the preimage of greg in g′. The diagram induces a map

preg : (SymX b)reg → g′reg := greg ×h//W

h

which is proper because (SymX b)reg → greg is proper and g′reg → greg is separated.Let us see now that it is also birational. Let gss denote the set of semi-simple elements in g; recall

that it is an open Zariski-dense set in g, and let greg,ss := greg ∩ gss be the set of elements which are both

regular and semi-simple. Let (SymX b)reg,ss denote the preimage of greg,ss in (SymX b)reg, and let hreg

and hreg//W denote the regular elements in h and h//W , respectively. Here, here we notice that hreg is

24This is a somewhat non-trivial result, so we are in some sense cheating by citing it here. However, it does not introduceany logical circularity, so we choose to omit the fairly computational proof to streamline our description of the geometric

situation.


simply the complement of the root hyperplanes in h, the W -action restricts to hreg and it makes senseto consider hreg//W . Then, for birationality, it suffices for us to show that

(18)

(SymX b)reg,ss- hreg

- h

greg,ss

?- hreg//W

?- h//W

?

is Cartesian. For this, we need only check that the left square is Cartesian. We claim that each ξ ∈ greg,ss

is contained in exactly |W | Borel subalgebras of g; indeed, because ξ ∈ greg,ss, its centralizer zg(ξ) is theunique Cartan subalgebra containing it. Therefore, the set of Borel subalgebras containing ξ is exactlythose which contain this Cartan subalgebra, of which there are |W | by definition. To specify one ofthese Borel subalgebras is to specify exactly one of the |W | possible images of ξ in the abstract Cartansubalgebra, hence we see that (18) is Cartesian and thus preg is birational.

Now, notice that the map g′reg → h is smooth as the pullback of the smooth map greg → h//W (byLemma 4.17) along the flat map h → h//W . Therefore, we see that g′reg is regular. Thus, by Zariski’smain theorem (see [Gro61, III.4.4.9]), it suffices for us to show that preg has finite fibers, for which it

suffices to check that preg is injective on tangent spaces. Indeed, note that the tangent space to SymX bat a point (b, ξ) with ξ ∈ b is given by g/b⊕ b and the map to the tangent space to g ×

h//Wh is given by

the map(µ1, µ2) 7→ (µ2, π([µ1, ξ]))

with π the projection b → h. Note here that this map is well defined because π vanishes on [b, b] = n.Thus, we must check that for µ1 ∈ g and ξ ∈ breg, if [µ1, ξ] ∈ n, then µ1 ∈ b. Indeed, if µ1 /∈ b, thenby the classification of Proposition 4.20, the lowest weight of [µ1, ξ] will be non-positive, meaning that[µ1, ξ] /∈ n, as needed.

Because greg is open in g, Lemma 4.18 implies that the induced map

p : SymX b→ g ×h//W

h

is birational. We now consider the following two diagrams.

SymX n - 0 N - 0

(a) (b)

SymX b

?- h?

g?

- h//W?

We claim that they commute and are Cartesian; for (a), this is clear, and for (b), this follows fromthe fact that nilpotent elements in g are exactly those which are killed by all functions in the image ofthe Chevalley homomorphism without a constant term. We may now fit all three diagrams together tosummarize the geometric situation.

(19)

SymX np - N

π - 0

(c) (d)

SymX b

h

?p- g ×

h//Wh

j

? φ′ - h

i

?

(e)

g? φ-

-

h//W?

Here, the maps of (cd) are given by those of (a), and the maps of (de) are given by those of (b). Thediagram commutes by construction; we now claim that each square is Cartesian. Indeed, for (d), thisfollows because (de) and (e) are both Cartesian, and for (c), this follows because (cd) and (d) are bothCartesian. We may now state Kostant’s theorem.


Theorem 4.19 (Kostant). The map p : SymX n→ N of (19) is a resolution of singularities, known asthe Springer resolution. Further, the map p satisfies p∗OSymX n ' ON .25

Before giving the proof, let us first see how it implies Proposition 4.14.

Proof of Proposition 4.14. By Theorem 4.19, the map p : SymX n→ N provides an isomorphism

(20) p∗OSymX n ' ON .

Recall from the definition of b that we had an exact sequence

0→ b→ g→ TX → 0.

Note that this sequence is a priori only left exact, but is right exact by Lemma 4.7. Therefore, we

see that TX ' g/b ' n−. Therefore, applying the identification given by the Killing form we haveT ∗X ' (n−)∗ ' n and hence

(21) SymX T∗X ' SymX n.

Further, the construction of the Chevalley isomorphism given in Theorem 3.13 provides an explicitdescription for N as the elements in g which vanish under every element of Sym(g∗)G without a constantterm; under the identification g ' g∗, this shows exactly that

(22) N ' Spec(Sym(g)/ Sym(g) · Sym(g)G+).

Combining the identifications (21) and (22), the isomorphism of (20) induces an isomorphism

Sym(g)/ Sym(g) · Sym(g)G+ ' Γ(N ,ON )→ Γ(SymX T∗X ,OSymX T∗X

) ' Γ(X,SymOX TX).

It remains now to check that this isomorphism is the map given in the statement of Proposition 4.14.Concretely, we must show that the diagram

SymX n ' SymX(n−)∗∼- SymX T

∗X

N?⊂ - g ' g∗

?

commutes, where the map SymX T∗X → g∗ is induced by differentiating the action of G on X. Indeed,

tracing through the identifications, this amounts to the fact that the double adjunction of a map g→ TXgives the map itself.

It remains now for us to prove Theorem 4.19. Our strategy will be to lift the resolution of singularities

SymX b→ g′ given by Lemma 4.18 to a resolution SymX n→ N . The remaining result will then followfrom a more detailed analysis of the regular elements in g. In particular, we will first classify all regularelements that lie within a specified Borel subalgebra b.

Proposition 4.20. An element ξ ∈ b is regular if and only if there exists a Cartan subalgebra h ⊂ b, asubset I ⊂ 1, . . . , n of the simple roots, and a decomposition

ξ = ξss +∑α∈R+

ξα

with ξss ∈ h and ξα ∈ gα such that the following properties hold:

• αi(ξss) = 0 for all i in I,• αi(ξss) 6= 0 for i /∈ I,• ξα = 0 if αi appears in the decomposition of α for i ∈ I, and• ξαi 6= 0 for i /∈ I.

Proof. Take ξ ∈ g and let ξ = ξss + ξn with ξss semi-simple, ξn nilpotent, and [ξss, ξn] = 0 be its Jordandecomposition. Write z = zg(ξss) and note that z is the direct sum of a semisimple Lie algebra and anabelian one with total rank rank g. Further, because zz(ξn) ⊂ zg(ξ) we see that ξ is regular if and onlyif ξn is regular inside the semisimple part of z.

25In fact, it is known that p is a rational resolution of singularities, meaning that Rp∗OSymX n ' ON in the derived

category. For k = C, Grauert-Riemenschneider vanishing (see [GR70, Satz 2.3]) shows that it suffices to check that the

canonical bundle of SymX n is trivial.


Now, suppose ξ ∈ b was regular. Pick a Borel subalgebra b′ ⊂ b of z containing ξ and a Cartansubalgebra h of z inside b′. Then h is also a Cartan subalgebra of g. Then, this determines a decomposition

b = h⊕⊕α∈R+

gα

along with a choice I ⊂ 1, . . . , n of a subset of the simple roots for which we may write ξ in the form

ξ = ξss +∑α∈R+

ξα

with ξα ∈ gα such that αi(ξss) = 0 for i ∈ I, αi(ξss) 6= 0 for i /∈ I, and ξα = 0 if αi for i ∈ I appears inthe decomposition of α as the sum of simple roots.

To complete the classification of regular elements, then it suffices to understand when nilpotent ele-ments of the form

∑α∈R+ ξα are regular. We claim that this occurs if and only if ξαi 6= 0 for all i /∈ I;

we omit the proof of this and instead refer the reader to the original in [Kos59, Theorem 5.3], whichproceeds by combinatorial considerations on principal sl2’s in g. Putting everything together, we haveobtained exactly the desired conditions on our decomposition.

We may now use Proposition 4.20 to analyze the codimension of g− greg in g.

Lemma 4.21. The codimension of g− greg in g is at least 2.

Proof. It suffices for us to show that for every Borel subalgebra b, the codimension of b− b ∩ (g− greg)is at least 2.26 Given a choice of b, we have a canonical projection

b→⊕α∈R+

gα

independent of the choice of h. Translating the result of Proposition 4.20, we see that the image of b−breg

under this projection is contained in the finite union of the hyperplanes ξαi 6= 0 and further that thefiber of b− breg above each point in

⊕α∈R+ gα has codimension at least 1, allowing us to conclude that

b− breg itself has codimension at least 2 in b, as needed.

Finally, before beginning the proof, we need one more auxiliary result, a standard lemma from com-mutative algebra.

Lemma 4.22. Consider a Cartesian diagram

X ′ ⊂ - Y ′ - Z ′

X?⊂ - Y

?- Z?

of affine schemes with Z ′ → Z flat and X → Y and X ′ → Y ′ closed embeddings. Then, if X hascodimension at least 2 in Y , X ′ has codimension at least 2 in Y ′.

Proof. Recall by construction that

OX′ = OX ⊗OZOZ′ .

Take a prime p ⊗ q of OX′ , where p and q are primes of OX and OZ′ which agree upon restriction toOZ . Because X has codimension at least 2 in Y , there exists a non-trivial chain of primes

p0 ⊂ p1 ⊂ p

in OX with corresponding preimages p′0 ⊂ p′1 ⊂ p′ in OZ . Now, OZ′ is flat over OZ , so we may applythe going down theorem to obtain a corresponding chain of primes

q0 ⊂ q1 ⊂ q

in OZ′ such that pi and qi both restrict to p′i. This gives a chain of primes pi ⊗ qi contained in p ⊗ qwhich shows that height(p⊗ q) ≥ 2, as needed.

26In fact, it is true that codim(g− greg) ≥ 3.


Proof of Theorem 4.19. Note that square (c) is Cartesian and p : SymX b → g′ is birational by Lemma4.18, so to show that p is birational, it suffices merely to check that the intersection of N and g′reg in g′

is non-empty. Indeed, the preimage of any regular nilpotent element of g lies in this intersection, so p isbirational.

Let us now show that p∗OSymX n ' ON . For this, we first consider the map p : SymX b→ g′. Denotethe composition

q : SymX b→ X × g→ g

of the closed embedding SymX b → X × g and the projection X × g → g. Note that q is projectivebecause X is a projective variety; therefore, p is projective because the projection g′ → g.

We now claim that g′ is normal. By Lemma 4.16, g′ is the pullback of a smooth variety along aflat morphism between smooth varieties, hence it is a local complete intersection and therefore Cohen-

Macaulay. Now, recall p : SymX b→ g′ is birational and restricts to an isomorphism to g′reg ⊂ g′, which

is regular because it is isomorphic to (SymX b)reg, which is smooth. Thus, to show that g′ is normal, itsuffices to check that g′ − g′reg has codimension at least 2. By Lemma 4.21, we know that g − greg hascodimension at least 2; because h→ h//W is flat by Lemma 4.16, by Lemma 4.22 we see that g′ − g′reg

has codimension 2 in g′. This implies that g′ is Cohen-Macaulay and regular in codimension 1, hencenormal. Therefore, by Zariski’s Main Theorem applied to the proper map p (see [Gro61, III.4.3.12]), wesee that p∗OSymX b ' Og′ .

We now claim that SymX b→ h is flat. Indeed, note that SymX b is smooth and equidimensional asa vector bundle over the flag variety, hence this is a map of Cohen-Macaulay schemes, so it suffices by[Gro65, IV.6.1.5] to check that each fiber is Cohen-Macaulay and has dimension dim g−dim h. But noticethat the fiber over a point ξ ∈ h is a vector bundle over X whose fiber over x ∈ X is µ ∈ bx | πx(µ) = ξfor πx : bx/nx → h the canonical projection. Thus, we see that each fiber is smooth of the correctdimension, so the map is flat as needed. Therefore, we may apply flat base change on (cd) and (d) tosee that

π∗p∗OSymX n ' i∗φ′∗p∗OSymX b ' i∗φ′∗p∗OSymX b ' π∗j

∗Og′ ' π∗ON .

But recall that N was affine, so this implies that p∗OSymX n ' ON .27

4.7. The global sections functor on the flag variety. In this subsection, we analyze the functorof global sections on the flag variety to prove Theorem 4.11. Let M be a DλX -module. For µ ∈ X anappropriately chosen dominant integral weight, we will instead consider M ⊗ Lµ. For this, let V−µ =Γ(X,Lµ) and V µ = Γ(X,Lµ)∗; by Theorem 3.9, V−µ and V µ are the (finite dimensional) irreduciblerepresentations of G of lowest weight −µ and highest weight µ, respectively. Denote by V−µ := OX⊗kV−µand Vµ := OX ⊗k V µ the corresponding sheaves.

Lemma 4.23. Let µ1, . . . , µk be a labeling of the set of weights of V µ so that µi < µj implies thati > j. Then, there are B-stable filtrations

0 = V0 ⊂ V1 ⊂ · · · ⊂ Vk = V µ

and

0 = W0 ⊂W1 ⊂ · · · ⊂Wk = V−µ,

such that Vi/Vi−1 is the one dimensional B-module corresponding to µi and Wi/Wi−1 is the one dimen-sional B-module corresponding to −µk−i+1.

Proof. Such a filtration of U(b)-modules is immediate from the classification of finite-dimensional rep-resentations of semisimple Lie algebras. Because G was connected and simply connected, it lifts to afiltration of B-modules.

Lemma 4.24. Let µ1, . . . , µk be a labeling of the multiset of weights of V µ so that µi < µj impliesthat i > j. Then, there are filtrations

0 = V0 ⊂ V1 ⊂ · · · ⊂ Vk = Vµ

and

0 =W0 ⊂ W1 ⊂ · · · ⊂ Wk = V−µof G-equivariant sheaves such that Vi/Vi−1 ' L−µi and Wi/Wi−1 ' Lµk−i+1 .

27Instead of passing through g′, we could have instead shown that N is normal by further analyzing the geometry of

G-orbits in N . Then, a direct application of Zariski’s main theorem to the birational map SymX n → N would give this

part of the result.


Proof. Construct these filtrations by applying the functor L(−) to the filtrations of Lemma 4.23. Theresult satisfies Vk := L(V µ) = Vµ and Wk := L(V−µ) = V−µ by Proposition 3.7 and Vi/Vi−1 ' L−µi

and Wi/Wi−1 ' Lµk−i+1 by Proposition 3.6(i).

Because µ is dominant, Lµ is globally generated, giving a natural surjection V−µ Lµ. Dualizing andtensoring by the line bundle Lµ, we obtain also an injection OX → Lµ ⊗ Vµ. Because the U(g)-modulestructure on Lµ is induced by the map g→ Γ(X,DµX), we see that these are maps of U(g)-modules. Byfunctoriality, we may tensor them with M to obtain maps of U(g)-modules

(23) M →M⊗ Lµ ⊗ Vµ

and

(24) M⊗ V−µ M⊗ Lµ.

Here (23) is evidently injective and (24) is surjective by right exactness of M ⊗ −. We now have thefollowing key lemma.

Lemma 4.25. Let µ be an integral weight and λ an arbitrary weight. For any DλX-module M, we havethe following:

(i) if λ is dominant, the map of (23) is a split injection of U(g)-modules, and(ii) if λ is regular, the map of (24) is a split surjection of U(g)-modules.

Proof. Pick a labeling µ1, . . . , µk of the weights of V µ; in particular, we note that µ1 = µ and µi < µfor i > 1. Consider now the corresponding filtrations given by Lemma 4.23. Viewing Vi and Wi asU(g)-modules via the G-equivariant structure, we obtain filtrations

0 = M⊗ Lµ ⊗ V0 ⊂M⊗ Lµ ⊗ V1 ⊂ · · · ⊂M⊗ Lµ ⊗ Vk = M⊗ Lµ ⊗ Vµ

and0 = M⊗W0 ⊂M⊗W1 ⊂ · · · ⊂M⊗Wk = M⊗ V−µ

of U(g)-modules, where for each i we have short exact sequences

(25) 0→M⊗ Lµ ⊗ Vi−1 →M⊗ Lµ ⊗ Vi →M⊗ Lµ ⊗ L−µi → 0

and

(26) 0→M⊗Wi−1 →M⊗Wi →M⊗ Lµk−i+1 → 0.

We claim now that for any local section z ∈ Z(g), the section

ai(z) :=

i∏i=1

(z − χ−λ−µ+µi(z))

acts by zero on M⊗ Lµ ⊗ Vi. For this, recall that by Proposition 4.13, Z(g) acts on M and Lµ via χ−λand χ−µ−ρ, respectively. Further, notice that if Z(g) acts on M1 and M2 via χµ1

and χµ2, then by the

definition of the U(g)-action on M1 ⊗M2, it acts on M1 ⊗M2 via χm1+µ2+ρ.28

The claim will now follow by induction on i. For the base case i = 1, Z(g) acts on M, Lµ, and V1

via χ−λ, χ−µ−ρ, and χ−µ1−ρ, respectively, hence it acts on M⊗ Lµ ⊗ V1 via χ−λ−µ+µ1 = χ−λ. For theinductive step, consider the short exact sequence (25); by the inductive hypothesis, ai−1(z) kills the firstterm, while Z(g)-acts by χλ−µ+µi on the last term, hence ai(z) kills the middle term, completing theinduction.

Taking i = k, we see that∏ki=1(z − χ−λ−µ+µi(z)) kills M⊗ Lµ ⊗ Vµ. By a similar argument, we see

that∏ki=1(z − χ−λ−µk−i+1

(z)) kills M ⊗ V−µ. Together, these show that the action of Z(g) is locallyfinite on M ⊗ Lµ ⊗ Vµ and M ⊗ V−µ, as the image of any section under the Z(g)-action is spanned bythe images of finite powers of the generators for Z(g).

Thus, M ⊗ Lµ ⊗ Vµ and M ⊗ V−µ split as U(g)-modules into the direct sum of generalized weight

spaces for Z(g) with weights parametrized by the weights of the U(b)-stable filtrations. To completethe proof of the lemma, for (i) it suffices to show that the inclusion (23) exhibits M as the kernel of theaction by (z − χ−λ(z))N for N large under the inclusion

0→M 'M⊗ Lµ ⊗ V1 →M⊗ Lµ ⊗ Vµ

and for (ii) that (24) exhibits M⊗Lµ as the quotient by the kernel of the action of∏ki=2(z−χ−λ−µi(z))N

for N large under the projection

M⊗ V−µ M⊗Wk/Wk−1 'M⊗ Lµ → 0.

28Here the addition of ρ is to compensate for the shift we introduced in the definition of the Harish-Chandra isomorphism.


For (i), we need to check that for µi < µ, we have χ−λ 6= χ−λ−µ+µi . By Corollary 3.17, it suffices tocheck that −λ and −λ− µ+ µi are not in the same W -orbit. Indeed, if

w(−λ) = −λ− µ+ µi,

we would have

w(λ)− λ = µ− µi > 0,

which is impossible because λ is dominant.For (ii), we need to check that for µi < µ, we have χ−λ−µi 6= χ−λ−µ. Again by Corollary 3.17, it

suffices to check that −λ− µi and −λ− µ are not in the same W -orbit; indeed, if

w(−λ− µi) = −λ− µ,

we would have

w(λ)− λ = µ− w(µi) ≥ 0

where the last inequality follows because −µi is a weight of the lowest weight module V−µ with lowestweight −µ. This implies that w(λ) = λ and hence that w = 1 because λ is regular. But this means thatµ = µi, a contradiction.29

Proof of Theorem 4.11. Let M be a DλX -module. For (i), we wish to show that Hk(X,M) = 0 for allk > 0. Recall that because cohomology commutes with direct limits, we have that

Hk(X,M) = lim−→Hk(X,F),

where we take the direct image over the directed set of coherent submodules F of M; it therefore sufficesfor us to show that Hk(X,F)→ Hk(X,M) is the zero map for all such F. Fix now a coherent submoduleF of M. Because Lµ is ample for µ regular by Proposition 3.8(iii), we may choose a regular integralweight µ so that

Hk(X,F ⊗ Lµ ⊗k

Γ(X,Lµ)∗) ' Hk(X,F ⊗ Lµ)⊗k

Γ(X,Lµ)∗ = 0

by Serre vanishing. Now, consider the following commutative diagram on the level of sheaves of abeliangroups.

F - M

F ⊗ Lµ ⊗k

Γ(X,Lµ)∗?

- M⊗ Lµ ⊗k

Γ(X,Lµ)∗?

Therefore, the resulting map Hk(X,F) → Hk(M ⊗ Lµ ⊗k

Γ(X,Lµ)∗) factors through Hk(F ⊗ Lµ ⊗k

Γ(X,Lµ)∗) = 0, so it is the zero map. On the other hand, by Lemma 4.25(i), we see that Hk(X,M)→Hk(M ⊗ Lµ ⊗

kΓ(X,Lµ)∗) is injective, meaning that Hk(X,F) → Hk(X,M) is zero for all coherent

submodules F of M, hence Hk(X,M) for k > 0, as desired.For (ii), suppose that Γ(X,M) = 0. We again pass to the coherent submodules F of M. For any such

F, we have an injection Γ(X,F) → Γ(X,M), hence Γ(X,F) = 0. Now, pick µ a regular integral weightso that F ⊗ Lµ is globally generated (again this is possible by Proposition 3.8(iii)). Then, by Lemma4.25(ii), we have a split surjection

F ⊗k

Γ(X,Lµ)→ F ⊗ Lµ,

which induces a surjection Γ(X,F) ⊗k

Γ(X,Lµ) Γ(X,F ⊗ Lµ) on global sections. Thus, we find that

Γ(X,F⊗Lµ) = 0, meaning that F⊗Lµ = 0 by global generation, hence F = 0. Therefore, every coherentsubmodule F of M is zero, so M = 0, as desired.

29It was crucial to the proof of (ii) that we could use regularity to conclude that w = 1, which is not necessary for (i).


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